Anti-Derivatives Exercise: 19.1 Class 11 Basic Mathematics Solution [NEB UPDATED]

Exercise 19.1

Find the indefinite integrals of:

1.(i)$\mathop \smallint \nolimits5{{\rm{x}}^3}dx$

Solution:

Or, $\mathop \smallint \nolimits5{{\rm{x}}^3}$ = 5$\mathop \smallint \nolimits{{\rm{x}}^3}$.dx = 5$.\frac{{{{\rm{x}}^{3 + 1}}}}{4}$ + c = $\frac{5}{4}$x⁴ + c.

 

(ii)$\mathop \smallint \nolimits7{{\rm{x}}^{\frac{5}{2}}}dx$

Solution:

Or, $\mathop \smallint \nolimits7{{\rm{x}}^{\frac{5}{2}}}$   = 7$.\frac{{{{\rm{x}}^{\frac{5}{2} + 1}}}}{{\frac{7}{2}}}$ + c = 2x^7/2 + c.

 

(iii)$\mathop \smallint \nolimits 2{{\rm{x}}^{\frac{{ - 7}}{2}}}dx$

Solution:

Or, $\mathop \smallint \nolimits 2{{\rm{x}}^{\frac{{ - 7}}{2}}}$ = 2$\mathop \smallint \nolimits {{\rm{x}}^{\frac{{ - 7}}{2}}}$.dx = 2$.\frac{{{{\rm{x}}^{ - 7/2 + 1}}}}{{ - 5/2}}$ + c = –$\frac{4}{5}$.x^–5/2 + c = $ - \frac{4}{{5{{\rm{x}}^{\frac{5}{2}}}}}$ + c.

 

(iv)$\mathop \smallint \nolimits4{{\rm{x}}^{ - 5}}$ .dx

Solution:

Or, $\mathop \smallint \nolimits4{{\rm{x}}^{ - 5}}$ .dx = 4.$\mathop \smallint \nolimits {{\rm{x}}^{ - 5}}$.dx = 4$.\frac{{{{\rm{x}}^{ - 5 + 1}}}}{{ - 4}}$ + c = – x^–4 + c = $ - \frac{1}{{{{\rm{x}}^4}}}$ + c.

 

2.(i)$\mathop \smallint \nolimits \left( {{{\rm{x}}^2} + 2} \right).{\rm{dx}}$

Solution:

Or, $\mathop \smallint \nolimits \left( {{{\rm{x}}^2} + 2} \right).{\rm{dx}}$  = $\frac{{{{\rm{x}}^{2 + 1}}}}{3}$ + 2x + c = $\frac{{{{\rm{x}}^3}}}{3}$ + 2x + c.

 

(ii)$\mathop \smallint \nolimits \left( {3{{\rm{x}}^2} + 2{\rm{x}} + 1} \right).{\rm{dx}}$ 

Solution:

Or, $\mathop \smallint \nolimits \left( {3{{\rm{x}}^2} + 2{\rm{x}} + 1} \right).{\rm{dx}}$ = $3\mathop \smallint \nolimits {{\rm{x}}^2}.{\rm{dx}} + 2.\mathop \smallint \nolimits {\rm{x}}.{\rm{dx}} + \mathop \smallint \nolimits1.{\rm{dx}}$ = 3.$\frac{{{{\rm{x}}^{2 + 1}}}}{3}$ + 2.$\frac{{{{\rm{x}}^{1 + 1}}}}{2}$ + x + c = x³ + x² + x + c.

 

(iii)$\mathop \smallint \nolimits \left( {{{\rm{x}}^{\frac{3}{4}}} + {{\rm{x}}^{\frac{1}{2}}} + 4{{\rm{x}}^{\frac{1}{3}}}} \right).{\rm{dx}}$

Solution:

$\mathop \smallint \nolimits \left( {{{\rm{x}}^{\frac{3}{4}}} + {{\rm{x}}^{\frac{1}{2}}} + 4{{\rm{x}}^{\frac{1}{3}}}} \right).{\rm{dx}}$ = $\mathop \smallint \nolimits {{\rm{x}}^{\frac{3}{4}}}.{\rm{dx}} + \mathop \smallint \nolimits {{\rm{x}}^{\frac{1}{2}}}.{\rm{dx}} + 4\mathop \smallint \nolimits {{\rm{x}}^{\frac{1}{3}}}.{\rm{dx}}$

= $\frac{{{{\rm{x}}^{\frac{3}{4} + 1}}}}{{\frac{7}{4}}} + \frac{{{{\rm{x}}^{\frac{1}{2} + 1}}}}{{\frac{3}{2}}} + 4.\frac{{{{\rm{x}}^{\frac{1}{3} + 1}}}}{{\frac{4}{3}}}$ + c.

= $\frac{4}{7}{{\rm{x}}^{\frac{7}{4}}} + \frac{2}{3}{{\rm{x}}^{\frac{3}{2}}} + 3{{\rm{x}}^{\frac{4}{3}}}$ + c.

 

(iv)$\mathop \smallint \nolimits \left( {2{\rm{x}} + 1} \right)\left( {3{\rm{x}} + 2} \right).{\rm{dx}}$ 

Solution:

$\mathop \smallint \nolimits \left( {2{\rm{x}} + 1} \right)\left( {3{\rm{x}} + 2} \right).{\rm{dx}}$ = $\mathop \smallint \nolimits \left( {6{{\rm{x}}^2} + 7{\rm{x}} + 2} \right).{\rm{dx}}$

= $6.\frac{{{{\rm{x}}^{2 + 1}}}}{3} + 7.\frac{{{{\rm{x}}^{1 + 1}}}}{2}$ + 2x + c.

= 2x³ + $\frac{7}{2}$x² + 2x + c.

 

(v)$\mathop \smallint \nolimits \left( {{{\rm{x}}^2} - \frac{1}{{{{\rm{x}}^2}}}} \right).{\rm{dx}}$

Solution:

$\mathop \smallint \nolimits \left( {{{\rm{x}}^2} - \frac{1}{{{{\rm{x}}^2}}}} \right).{\rm{dx}}$ = $\mathop \smallint \nolimits {{\rm{x}}^2}.{\rm{dx}} - $$\mathop \smallint \nolimits \frac{1}{{{{\rm{x}}^2}}}.{\rm{dx}}$

= $\mathop \smallint \nolimits {{\rm{x}}^2}.{\rm{dx}} - \mathop \smallint \nolimits {{\rm{x}}^{ - 2}}.{\rm{dx}}$ = $\frac{{{{\rm{x}}^{2 + 1}}}}{3} - \frac{{{{\rm{x}}^{ - 2 + 1}}}}{{ - 1}}$ + c.

= $\frac{1}{3}$x³ + x^–1 + c = $\frac{1}{3}$x³ + $\frac{1}{{\rm{x}}}$ + c.

 

(vi)$\mathop \smallint \nolimits \left( {\sqrt {\rm{x}}  - \frac{1}{{\sqrt {\rm{x}} }}} \right).{\rm{dx}}$

Solution:

$\mathop \smallint \nolimits \left( {\sqrt {\rm{x}}  - \frac{1}{{\sqrt {\rm{x}} }}} \right).{\rm{dx}}$ = $\mathop \smallint \nolimits \left( {{{\rm{x}}^{\frac{1}{2}}} - \frac{1}{{{{\rm{x}}^{\frac{1}{2}}}}}} \right).{\rm{dx}}$ = $\mathop \smallint \nolimits {{\rm{x}}^{\frac{1}{2}}}.{\rm{dx}} - $$\mathop \smallint \nolimits {{\rm{x}}^{ - \frac{1}{2}}}.{\rm{dx}}$

= $\frac{{{{\rm{x}}^{\frac{1}{2} + 1}}}}{{\frac{3}{2}}} - \frac{{{{\rm{x}}^{ - \frac{1}{2} + 1}}}}{{\frac{1}{2}}}$ + c = $\frac{2}{3}$.x^3/2 – 2x^1/2 + c.

 

(vii)$\mathop \smallint \nolimits \sqrt {\rm{x}} .\left( {{{\rm{x}}^2} - 5} \right).{\rm{dx}}$ 

Solution:

$\mathop \smallint \nolimits \sqrt {\rm{x}} .\left( {{{\rm{x}}^2} - 5} \right).{\rm{dx}}$ = $\mathop \smallint \nolimits {{\rm{x}}^{\frac{1}{2}}}.\left( {{{\rm{x}}^2} - 5} \right).{\rm{dx}}$ = $\mathop \smallint \nolimits \left( {{{\rm{x}}^{\frac{5}{2}}} - 5{{\rm{x}}^{\frac{1}{2}}}} \right).{\rm{dx}}$

= $\mathop \smallint \nolimits {{\rm{x}}^{\frac{5}{2}}}.{\rm{dx}} - 5\mathop \smallint \nolimits {{\rm{x}}^{\frac{1}{2}}}.{\rm{dx}}$

= $\frac{{{{\rm{x}}^{\frac{5}{2} + 1}}}}{{\frac{7}{2}}} - \frac{{5{{\rm{x}}^{\frac{3}{2}}}}}{{\frac{3}{2}}}$ + c = $\frac{2}{7}$.x^7/2 – $\frac{{10}}{3}$.x^3/2 + c.

 

(viii)$\mathop \smallint \nolimits {\left( {{\rm{x}} - 3} \right)^2}.{\rm{dx}}$ 

Solution:

$\mathop \smallint \nolimits {\left( {{\rm{x}} - 3} \right)^2}.{\rm{dx}}$ = $\mathop \smallint \nolimits \left( {{{\rm{x}}^2} - 6{\rm{x}} + 9} \right).{\rm{dx}}$ = $\mathop \smallint \nolimits {{\rm{x}}^2}.{\rm{dx}} - 6\mathop \smallint \nolimits {\rm{x}}.{\rm{dx}} + 9\mathop \smallint \nolimits 1.{\rm{dx}}$

= $\frac{{{{\rm{x}}^3}}}{3} - \frac{{6{{\rm{x}}^2}}}{2}$ + 9x + c = $\frac{1}{3}$x³ – 3x² + 9x + c.

 

(ix)$\mathop \smallint \nolimits \frac{{3{{\rm{x}}^2} - 5{\rm{x}} + 2}}{{\rm{x}}}.{\rm{dx}}$ 

Solution:

$\mathop \smallint \nolimits \frac{{3{{\rm{x}}^2} - 5{\rm{x}} + 2}}{{\rm{x}}}.{\rm{dx}}$ = $\mathop \smallint \nolimits \left( {\frac{{3{{\rm{x}}^2}}}{{\rm{x}}} - \frac{{5{\rm{x}}}}{{\rm{x}}} + \frac{2}{{\rm{x}}}} \right).{\rm{dx}}$ = 3.$\mathop \smallint \nolimits {\rm{x}}.{\rm{dx}} - 5\mathop \smallint \nolimits {\rm{dx}} + 2\mathop \smallint \nolimits \frac{1}{{\rm{x}}}.{\rm{dx}}$

= 3.$\frac{{{{\rm{x}}^2}}}{2}$ – 5x + 2.logx + c.

(x)$\mathop \smallint \nolimits \frac{{{\rm{a}}{{\rm{x}}^2} + {\rm{bx}} + {\rm{c}}}}{{{{\rm{x}}^2}}}.{\rm{dx}}$

Solution:

$\mathop \smallint \nolimits \frac{{{\rm{a}}{{\rm{x}}^2} + {\rm{bx}} + {\rm{c}}}}{{{{\rm{x}}^2}}}.{\rm{dx}}$ = $\mathop \smallint \nolimits \left( {\frac{{{\rm{a}}{{\rm{x}}^2}}}{{{{\rm{x}}^2}}} + \frac{{{\rm{bx}}}}{{{{\rm{x}}^2}}} + \frac{{\rm{c}}}{{{{\rm{x}}^2}}}} \right).{\rm{dx}}$ = $\mathop \smallint \nolimits \left( {{\rm{a}} + \frac{{\rm{b}}}{{\rm{x}}} + \frac{{\rm{c}}}{{{{\rm{x}}^2}}}} \right)$.dx

= a$\mathop \smallint \nolimits 1.{\rm{dx}}$ + b$\mathop \smallint \nolimits \frac{1}{{\rm{x}}}.{\rm{dx}}$ + c $\mathop \smallint \nolimits {{\rm{x}}^{ - 2}}.{\rm{dx}}$

= ax + b.logx + $\frac{{{\rm{c}}{{\rm{x}}^{ - 2 + 1}}}}{{ - 1}}$ + k

= ax + b.logx – $\frac{{\rm{c}}}{{\rm{x}}}$ + k.

 

(xi)$\mathop \smallint \nolimits \left( {{{\rm{x}}^2} + 3{\rm{x}} + 5} \right).{{\rm{x}}^{ - \frac{1}{3}}}.{\rm{dx}}$ 

Solution:

$\mathop \smallint \nolimits \left( {{{\rm{x}}^2} + 3{\rm{x}} + 5} \right).{{\rm{x}}^{ - \frac{1}{3}}}.{\rm{dx}}$ = $\mathop \smallint \nolimits \left( {{{\rm{x}}^{2 - \frac{1}{3}}} + 3{{\rm{x}}^{1 - \frac{1}{3}}} + 5{{\rm{x}}^{ - \frac{1}{3}}}} \right).{\rm{dx}}$ = $\mathop \smallint \nolimits ({{\rm{x}}^{\frac{5}{3}}} + 3{{\rm{x}}^{\frac{2}{3}}} + 5{{\rm{x}}^{ - \frac{1}{3}}}){\rm{\: }}$.dx

=  $\mathop \smallint \nolimits {{\rm{x}}^{\frac{5}{3}}}.{\rm{dx}}$ + 3$\mathop \smallint \nolimits {{\rm{x}}^{\frac{2}{3}}}.{\rm{dx}}$ + 5$\mathop \smallint \nolimits {{\rm{x}}^{ - \frac{1}{3}}}.{\rm{dx}}$

= $\frac{{{{\rm{x}}^{\frac{5}{3} + 1}}}}{{\frac{8}{3}}} + \frac{{3{{\rm{x}}^{\frac{2}{3} + 1}}}}{{\frac{5}{3}}} + \frac{{5{{\rm{x}}^{ - \frac{1}{3} + 1}}}}{{\frac{2}{3}}}$ + c

= $\frac{3}{8}$.x^8/3 + $\frac{9}{5}$.x^5/3 + $\frac{{15}}{2}$.x^2/3 + c.

 

3.(i)$\mathop \smallint \nolimits {\left( {3{\rm{x}} + 5} \right)^4}.{\rm{dx}}$

Solution:

Or, $\mathop \smallint \nolimits {\left( {3{\rm{x}} + 5} \right)^4}.{\rm{dx}}$ = $\frac{{{{\left( {3{\rm{x}} + 5} \right)}^{4 + 1}}}}{{5.3}}$ + c $\frac{1}{{15}}$(3x + 5)⁵ + c.

 

(ii)$\mathop \smallint \nolimits {\left( {{\rm{a}} - {\rm{bx}}} \right)^5}.{\rm{dx}}$

Solution:

Or, $\mathop \smallint \nolimits {\left( {{\rm{a}} - {\rm{bx}}} \right)^5}.{\rm{dx}}$ = $\frac{{{{\left( {{\rm{a}} - {\rm{bx}}} \right)}^{5 + 1}}}}{{6.\left( { - {\rm{b}}} \right)}}$ + c =$ - \frac{1}{{6{\rm{b}}}}$(a – bx)⁶ + c.

 

(iii)$\mathop \smallint \nolimits {\left( {{\rm{c}} + {\rm{dx}}} \right)^{ - \frac{3}{2}}}.{\rm{dx}}$

Solution:

Or, $\mathop \smallint \nolimits {\left( {{\rm{c}} + {\rm{dx}}} \right)^{ - \frac{3}{2}}}.{\rm{dx}}$ = $\frac{{{{\left( {{\rm{c}} + {\rm{dx}}} \right)}^{ - \frac{3}{2} + 1}}}}{{ - \frac{1}{2}.{\rm{d}}}}$ + k = $ - \frac{2}{{\rm{d}}}$(c + dx)^–1/2 + k.

 

(iv)$\mathop \smallint \nolimits \frac{1}{{\sqrt {2{\rm{x}} + 7} }}.{\rm{dx}}$

Solution:

Or, $\mathop \smallint \nolimits \frac{1}{{\sqrt {2{\rm{x}} + 7} }}.{\rm{dx}}$ = $\mathop \smallint \nolimits \frac{1}{{{{\left( {2{\rm{x}} + 7} \right)}^{\frac{1}{2}}}}}.{\rm{dx}}$ = $\mathop \smallint \nolimits {\left( {2{\rm{x}} + 1} \right)^{ - \frac{1}{2}}}.{\rm{dx}}$

= $\frac{{{{\left( {2{\rm{x}} + 7} \right)}^{ - \frac{1}{2} + 1}}}}{{\frac{1}{2}.2}}$ + c = $\sqrt {2{\rm{x}} + 7} $ + c.

 

(v)$\mathop \smallint \nolimits [{\rm{x}} + \frac{1}{{{{\left( {{\rm{x}} + 3} \right)}^2}}}.{\rm{dx}}$

Solution:

Or, $\mathop \smallint \nolimits [{\rm{x}} + \frac{1}{{{{\left( {{\rm{x}} + 3} \right)}^2}}}.{\rm{dx}}$ = $\mathop \smallint \nolimits {\rm{x}}.{\rm{dx}}$ + $\mathop \smallint \nolimits \frac{1}{{{{\left( {{\rm{x}} + 3} \right)}^2}}}.{\rm{dx}}$

= $\mathop \smallint \nolimits {\rm{x}}.{\rm{dx}}$ + $\mathop \smallint \nolimits {\left( {{\rm{x}} + 3} \right)^{ - 2}}.{\rm{dx}}$

= $\frac{{{{\rm{x}}^2}}}{2}$ + $\frac{{{{\left( {{\rm{x}} + 3} \right)}^{ - 2 + 1}}}}{{ - 1}}$ + c = $\frac{{{{\rm{x}}^2}}}{2} - {\left( {{\rm{x}} + 3} \right)^{ - 1}}$ + c.

= $\frac{{{{\rm{x}}^2}}}{2} - \frac{1}{{{\rm{x}} + 3}} + {\rm{c}}$.

 

(vi) $\mathop \smallint \nolimits \left[ {4 + \frac{1}{{{{\left( {5{\rm{x}} + 1} \right)}^2}}}} \right].{\rm{dx}}$ 

Solution:

Or, $\mathop \smallint \nolimits \left[ {4 + \frac{1}{{{{\left( {5{\rm{x}} + 1} \right)}^2}}}} \right].{\rm{dx}}$ = 4.$\mathop \smallint \nolimits 1.{\rm{dx}}$ + $\mathop \smallint \nolimits \frac{1}{{{{\left( {5{\rm{x}} + 1} \right)}^2}}}.{\rm{dx}}$

= 4x + $\frac{{{{\left( {5{\rm{x}} + 1} \right)}^{ - 2 + 1}}}}{{ - \frac{1}{5}}}$ + c = 4x – $\frac{1}{{5\left( {5{\rm{x}} + 1} \right)}}$ + c. 

 

(vii)$\mathop \smallint \nolimits \frac{{{\rm{x}} + 3}}{{{\rm{x}} - 3}}.{\rm{dx}}$

Solution:

Or, $\mathop \smallint \nolimits \frac{{{\rm{x}} + 3}}{{{\rm{x}} - 3}}.{\rm{dx}}$ = $\mathop \smallint \nolimits \frac{{\left( {{\rm{x}} - 3} \right) + 6}}{{{\rm{x}} - 3}}.{\rm{dx}}$

= $\mathop \smallint \nolimits \left( {\frac{{{\rm{x}} - 3}}{{{\rm{x}} - 3}} + \frac{6}{{{\rm{x}} - 3}}} \right).{\rm{dx}}$

= $\mathop \smallint \nolimits 1.{\rm{dx}} + 6$

= $\mathop \smallint \nolimits \frac{1}{{{\rm{x}} - 3}}.{\rm{dx}}$

= x + 6.log(x – 3) + c.

 

(viii) $\mathop \smallint \nolimits \frac{{3{\rm{x}} - 1}}{{{\rm{x}} - 2}}.{\rm{dx}}$ 

Or, $\mathop \smallint \nolimits \frac{{3{\rm{x}} - 1}}{{{\rm{x}} - 2}}.{\rm{dx}}$ = $\mathop \smallint \nolimits \frac{{3{\rm{x}} - 6 + 5}}{{{\rm{x}} - 2}}.{\rm{dx}}$

= $\mathop \smallint \nolimits \left[ {\frac{{3\left( {{\rm{x}} - 2} \right)}}{{{\rm{x}} - 2}} + \frac{5}{{{\rm{x}} - 2}}} \right].{\rm{dx}}$

= 3.$\mathop \smallint \nolimits {\rm{dx}} + 5$

= $\mathop \smallint \nolimits \frac{1}{{{\rm{x}} - 2}}.{\rm{dx}}$

= 3x + 5.log(x – 2) + c.

 

(ix)$\mathop \smallint \nolimits \frac{{{{\rm{x}}^2} + 3{\rm{x}} + 3}}{{{\rm{x}} + 1}}.{\rm{dx}}$ 

Solution:

Or, $\mathop \smallint \nolimits \frac{{{{\rm{x}}^2} + 3{\rm{x}} + 3}}{{{\rm{x}} + 1}}.{\rm{dx}}$ = $\mathop \smallint \nolimits \frac{{{\rm{x}}\left( {{\rm{x}} + 1} \right) + 2\left( {{\rm{x}} + 1} \right) + 1}}{{{\rm{x}} + 1}}.{\rm{dx}}$

= $\mathop \smallint \nolimits \frac{{\left( {{\rm{x}} + 1} \right)\left( {{\rm{x}} + 2} \right) + 1}}{{{\rm{x}} + 1}}.{\rm{dx}}$

= $\mathop \smallint \nolimits \left( {{\rm{x}} + 2 + \frac{1}{{{\rm{x}} + 1}}} \right).{\rm{dx}}$

= $\mathop \smallint \nolimits {\rm{x}}.{\rm{dx}} + 2\mathop \smallint \nolimits 1.{\rm{dx}} + \mathop \smallint \nolimits \frac{1}{{{\rm{x}} + 1}}.{\rm{dx}}$

= $\frac{{{{\rm{x}}^2}}}{2}$ + 2x + log(x + 1) + c.

 

(x)$\mathop \smallint \nolimits \frac{{{{\rm{x}}^2} + 5}}{{{\rm{x}} + 2}}.{\rm{dx}}$

Solution:

Or, $\mathop \smallint \nolimits \frac{{{{\rm{x}}^2} + 5}}{{{\rm{x}} + 2}}.{\rm{dx}}$ = $\mathop \smallint \nolimits \frac{{{{\rm{x}}^2} - 4 + 9}}{{{\rm{x}} + 2}}.{\rm{dx}}$

= $\mathop \smallint \nolimits \frac{{\left( {{\rm{x}} + 2} \right)\left( {{\rm{x}} - 2} \right) + 9}}{{{\rm{x}} + 2}}.{\rm{dx}}$

= $\mathop \smallint \nolimits \left( {{\rm{x}} - 2 + \frac{9}{{{\rm{x}} + 2}}} \right).{\rm{dx}}$

= $\mathop \smallint \nolimits {\rm{x}}.{\rm{dx}} - 2\mathop \smallint \nolimits 1.{\rm{dx}} + 9\mathop \smallint \nolimits \frac{1}{{{\rm{x}} + 2}}.{\rm{dx}}$

= $\frac{{{{\rm{x}}^2}}}{2}$ – 2x + 9log(x + 2) – c.

 

4.(i)$\mathop \smallint \nolimits {\rm{x}}\sqrt {{\rm{x}} + 1} .{\rm{dx}}$

Solution:

Or, $\mathop \smallint \nolimits {\rm{x}}\sqrt {{\rm{x}} + 1} .{\rm{dx}}$ = $\mathop \smallint \nolimits \left\{ {\left( {{\rm{x}} + 1} \right) - 1} \right\}{\left\{ {{\rm{x}} + 1} \right\}^{\frac{1}{2}}}.{\rm{dx}}$

= $\mathop \smallint \nolimits \left\{ {{{\left( {{\rm{x}} + 1} \right)}^{\frac{3}{2}}} - {{\left( {{\rm{x}} + 1} \right)}^{\frac{1}{2}}}} \right\}.{\rm{dx}}$

= $\mathop \smallint \nolimits {\left( {{\rm{x}} + 1} \right)^{\frac{3}{2}}}.{\rm{dx}}$ – $\mathop \smallint \nolimits {\left( {{\rm{x}} + 1} \right)^{\frac{1}{2}}}.{\rm{dx}}$

= $\frac{{{{\left( {{\rm{x}} + 1} \right)}^{\frac{5}{2}}}}}{{\frac{5}{2}}}$ – $\frac{{{{\left( {{\rm{x}} + 1} \right)}^{\frac{3}{2}}}}}{{\frac{3}{2}}}$ + c.

= $\frac{2}{5}$(x + 1)^5/2 – $\frac{2}{3}$(x + 1)^3/2 + c.

  

(ii)$\mathop \smallint \nolimits {\rm{x}}\sqrt {{\rm{ax}} + {\rm{b}}} .{\rm{dx}}$ 

Solution:

Or, $\mathop \smallint \nolimits {\rm{x}}\sqrt {{\rm{ax}} + {\rm{b}}} .{\rm{dx}}$ = $\frac{1}{{\rm{a}}}\mathop \smallint \nolimits {\rm{ax}}\sqrt {{\rm{ax}} + {\rm{b}}} .{\rm{dx}}$

= $\frac{1}{{\rm{a}}}\mathop \smallint \nolimits \left\{ {\left( {{\rm{ax}} + {\rm{b}}} \right) - {\rm{b}}} \right\}{\left( {{\rm{ax}} + {\rm{b}}} \right)^{\frac{1}{2}}}.{\rm{dx}}$

= $\frac{1}{{\rm{a}}}\mathop \smallint \nolimits \left\{ {{{\left( {{\rm{ax}} + {\rm{b}}} \right)}^{\frac{3}{2}}} - {\rm{b}}{{\left( {{\rm{ax}} + {\rm{b}}} \right)}^{\frac{1}{2}}}} \right\}.{\rm{dx}}$

= $\frac{1}{{\rm{a}}}\mathop \smallint \nolimits {\left( {{\rm{ax}} + {\rm{b}}} \right)^{\frac{3}{2}}}.{\rm{dx}} - \frac{{\rm{b}}}{{\rm{a}}}\mathop \smallint \nolimits {\left( {{\rm{ax}} + {\rm{b}}} \right)^{\frac{1}{2}}}.{\rm{dx}}$

= $\frac{1}{{\rm{a}}}\frac{{\left( {{{\left( {{\rm{ax}} + {\rm{b}}} \right)}^{\frac{5}{2}}}} \right)}}{{\frac{5}{2}.{\rm{a}}}} - \frac{{\frac{{\rm{b}}}{{\rm{a}}}\left( {{{\left( {{\rm{ax}} + {\rm{b}}} \right)}^{\frac{3}{2}}}} \right)}}{{\frac{3}{2}.{\rm{a}}}}$ + c.

= $\frac{1}{{{{\rm{a}}^2}}}\left[ {\frac{2}{5}{{\left( {{\rm{ax}} + {\rm{b}}} \right)}^{\frac{5}{2}}} - \frac{{2{\rm{b}}}}{3}{{\left( {{\rm{ax}} + {\rm{b}}} \right)}^{\frac{3}{2}}}} \right]$ + c.

 

(iii)$\mathop \smallint \nolimits 2{\rm{x}}\sqrt {2{\rm{x}} + 3} .{\rm{dx}}$

Solution:

Or, $\mathop \smallint \nolimits 2{\rm{x}}\sqrt {2{\rm{x}} + 3} .{\rm{dx}}$

= $\mathop \smallint \nolimits \left\{ {{{\left( {2{\rm{x}} + 3} \right)}^{\frac{3}{2}}} - 3{{\left( {2{\rm{x}} + 3} \right)}^{\frac{1}{2}}}} \right\}.{\rm{dx}}$

= $\mathop \smallint \nolimits {\left( {2{\rm{x}} + 3} \right)^{\frac{3}{2}}}.{\rm{dx}} - 3\mathop \smallint \nolimits {\left( {2{\rm{x}} + 3} \right)^{\frac{1}{2}}}.{\rm{dx}}$

= $\frac{{{{\left( {2{\rm{x}} + 3} \right)}^{\frac{5}{2}}}}}{{\frac{5}{2}.2}} - \frac{{3\left( {{{\left( {2{\rm{x}} + 3} \right)}^{\frac{3}{2}}}} \right)}}{{\frac{3}{2}.2}}$ + c.

= $\frac{1}{5}$(2x + 3)^5/2 – (2x + 3)^3/2 + c.

 

(iv)$\mathop \smallint \nolimits 5{\rm{x}}\sqrt {5{\rm{x}} + 2} .{\rm{dx}}$

Solution:

Or, $\mathop \smallint \nolimits 5{\rm{x}}\sqrt {5{\rm{x}} + 2} .{\rm{dx}}$

= $\mathop \smallint \nolimits {\left\{ {\left( {5{\rm{x}} + 2} \right) - 2} \right\}^{\frac{3}{2}}} - 2{\left( {5{\rm{x}} + 2} \right)^{\frac{1}{2}}}\} .{\rm{dx}}$

= $\mathop \smallint \nolimits {\left( {5{\rm{x}} + 2} \right)^{\frac{3}{2}}}.{\rm{dx}} - 2\mathop \smallint \nolimits {\left( {5{\rm{x}} + 2} \right)^{\frac{1}{2}}}.{\rm{dx}}$

= $\frac{{{{\left( {5{\rm{x}} + 2} \right)}^{\frac{5}{2}}}}}{{\frac{5}{2}.2}} - 2.\frac{{{{\left( {5{\rm{x}} + 2} \right)}^{\frac{3}{2}}}}}{{\frac{3}{2}.5}}$ + c.

= $\frac{2}{{25}}$(5x + 2)^5/2 – $\frac{4}{{15}}$(5x + 2)^3/2 + c.

 

(v)$\mathop \smallint \nolimits \left( {{\rm{x}} + 2} \right)\sqrt {3{\rm{x}} + 2} .{\rm{dx}}$

Solution:

Or, $\mathop \smallint \nolimits \left( {{\rm{x}} + 2} \right)\sqrt {3{\rm{x}} + 2} .{\rm{dx}}$

= $\frac{1}{3}\mathop \smallint \nolimits \left\{ {\left( {3{\rm{x}} + 2} \right) + 4} \right\}{\left( {3{\rm{x}} + 2} \right)^{\frac{1}{2}}}.{\rm{dx}}$

= $\frac{1}{3}\mathop \smallint \nolimits {\left( {3{\rm{x}} + 2} \right)^{\frac{3}{2}}}.{\rm{\: }} + 4{\left( {3{\rm{x}} + 2} \right)^{\frac{1}{2}}}.{\rm{dx}}$

=  $\frac{1}{3}\mathop \smallint \nolimits {\left( {3{\rm{x}} + 2} \right)^{\frac{3}{2}}}.{\rm{\: }} + \frac{4}{3}{\rm{\: }}\mathop \smallint \nolimits {\left( {3{\rm{x}} + 2} \right)^{\frac{1}{2}}}.{\rm{dx}}$

= $\frac{1}{3}$$\frac{{{{\left( {3{\rm{x}} + 2} \right)}^{\frac{5}{2}}}}}{{\frac{5}{2}.3}}$ + $\frac{4}{3}$$\frac{{{{\left( {3{\rm{x}} + 2} \right)}^{\frac{3}{2}}}}}{{\frac{3}{2}.3}}$ + c.

= $\frac{2}{{45}}$ (3x + 2)^5/2 + $\frac{8}{{27}}$ (3x + 2)^3/2 + c

= $\frac{2}{9}$$\left[ {\frac{1}{5}{{\left( {3{\rm{x}} + 2} \right)}^{\frac{5}{2}}} + \frac{4}{3}{{\left( {3{\rm{x}} + 2} \right)}^{\frac{3}{2}}}} \right]$ + c.

  

(vi) $\mathop \smallint \nolimits \left( {2{\rm{x}} + 3} \right)\sqrt {3{\rm{x}} + 1} .{\rm{dx}}$

Solution:

Or, $\mathop \smallint \nolimits \left( {2{\rm{x}} + 3} \right)\sqrt {3{\rm{x}} + 1} .{\rm{dx}}$

= $\frac{1}{3}\mathop \smallint \nolimits \left( {6{\rm{x}} + 9} \right)\sqrt {3{\rm{x}} + 1} .{\rm{dx}}$

= $\frac{1}{3}{\rm{\: }}\mathop \smallint \nolimits \left\{ {\left( {6{\rm{x}} + 2} \right) + 7} \right\}.{\left( {3{\rm{x}} + 1} \right)^{\frac{1}{2}}}.{\rm{dx}}$

=  $\frac{1}{3}$$\mathop \smallint \nolimits \{ {\rm{(}}2\left( {3{\rm{x}} + 1} \right) + 7{\rm{\} }}{\left( {3{\rm{x}} + 1} \right)^{\frac{1}{2}}}.{\rm{dx}}$

= $\frac{2}{3}$$\mathop \smallint \nolimits {\left( {3{\rm{x}} + 1} \right)^{\frac{3}{2}}}.{\rm{dx}}$ + $\frac{7}{3}$$\mathop \smallint \nolimits {\left( {3{\rm{x}} + 1} \right)^{\frac{1}{2}}}$.dx

= $\frac{2}{3}$$\frac{{{{\left( {3{\rm{x}} + 1} \right)}^{\frac{5}{2}}}}}{{\frac{5}{2}.3}}$ + $\frac{7}{3}$$\frac{{{{\left( {3{\rm{x}} + 1} \right)}^{\frac{3}{2}}}}}{{\frac{3}{2}.3}}$ + c

= $\frac{4}{{45}}$ (3x + 1)^5/2 + $\frac{{14}}{{27}}$ (3x + 1)^3/2 + c

= $\frac{2}{9}$$\left[ {\frac{2}{5}{{\left( {3{\rm{x}} + 1} \right)}^{\frac{5}{2}}} + \frac{7}{3}{{\left( {3{\rm{x}} + 1} \right)}^{\frac{3}{2}}}} \right]$ + c.

 

(vii)$\mathop \smallint \nolimits \left( {5{\rm{x}} + 3} \right)\sqrt {4{\rm{x}} + 1} .{\rm{dx}}$

Solution:

$\mathop \smallint \nolimits \left( {5{\rm{x}} + 3} \right)\sqrt {4{\rm{x}} + 1} .{\rm{dx}}$

= $\frac{1}{4}\mathop \smallint \nolimits \left( {20{\rm{x}} + 12} \right)\sqrt {4{\rm{x}} + 1} .{\rm{dx}}$

= $\frac{1}{4}{\rm{\: }}\mathop \smallint \nolimits \left\{ {\left( {20{\rm{x}} + 5} \right) + 7} \right\}.{\left( {4{\rm{x}} + 1} \right)^{\frac{1}{2}}}.{\rm{dx}}$

=  $\frac{1}{4}$$\mathop \smallint \nolimits \{ {\rm{(}}5\left( {4{\rm{x}} + 1} \right) + 7{\rm{\} }}{\left( {4{\rm{x}} + 1} \right)^{\frac{1}{2}}}.{\rm{dx}}$

= $\frac{5}{4}$$\mathop \smallint \nolimits {\left( {4{\rm{x}} + 1} \right)^{\frac{3}{2}}}.{\rm{dx}}$ + $\frac{7}{4}$$\mathop \smallint \nolimits {\left( {4{\rm{x}} + 1} \right)^{\frac{1}{2}}}$.dx

= $\frac{5}{4}$$\frac{{{{\left( {4{\rm{x}} + 1} \right)}^{\frac{5}{2}}}}}{{\frac{5}{2}.4}}$ + $\frac{7}{3}$$\frac{{{{\left( {4{\rm{x}} + 1} \right)}^{\frac{3}{2}}}}}{{\frac{3}{2}.4}}$ + c

= $\frac{1}{8}$ (4x + 1)^5/2 + $\frac{7}{{24}}$ (4x + 1)^3/2 + c

= $\frac{1}{8}$$\left[ {{{\left( {4{\rm{x}} + 1} \right)}^{\frac{5}{2}}} + \frac{7}{3}{{\left( {4{\rm{x}} + 1} \right)}^{\frac{3}{2}}}} \right]$ + c.

 

(viii)$\mathop \smallint \nolimits \frac{{{\rm{x}} + 2}}{{\sqrt {{\rm{x}} + 1} }}.{\rm{dx}}$ 

Solution:

$\mathop \smallint \nolimits \frac{{{\rm{x}} + 2}}{{\sqrt {{\rm{x}} + 1} }}.{\rm{dx}}$ = $\mathop \smallint \nolimits \frac{{\left( {{\rm{x}} + 1} \right) + 1}}{{{{\left( {{\rm{x}} + 1} \right)}^{\frac{1}{2}}}}}.{\rm{dx}}$

=$\mathop \smallint \nolimits \left\{ {\frac{{{\rm{x}} + 1}}{{{{\left( {{\rm{x}} + 1} \right)}^{\frac{1}{2}}}}} + \frac{1}{{{{\left( {{\rm{x}} + 1} \right)}^{\frac{1}{2}}}}}} \right\}.{\rm{dx}}$

= $\mathop \smallint \nolimits \left\{ {{{\left( {{\rm{x}} + 1} \right)}^{\frac{1}{2}}} + {{\left( {{\rm{x}} + 1} \right)}^{ - \frac{1}{2}}}} \right\}.{\rm{dx}}$

=   $\mathop \smallint \nolimits {\left( {{\rm{x}} + 1} \right)^{\frac{1}{2}}}.{\rm{dx}} + $$\mathop \smallint \nolimits {\left( {{\rm{x}} + 1} \right)^{ - \frac{1}{2}}}$.dx

= $\frac{{{{\left( {{\rm{x}} + 1} \right)}^{\frac{3}{2}}}}}{{\frac{3}{2}}} + \frac{{{{\left( {{\rm{x}} + 2} \right)}^{ - \frac{1}{2}}}}}{{\frac{1}{2}}}$ + c

= $\frac{2}{3}$ (x + 1)^3/2 + 2(x + 1)^1/2 + c.

  

(ix)$\mathop \smallint \nolimits \frac{{3{\rm{x}} + 4}}{{\sqrt {{\rm{x}} + 1} }}.{\rm{dx}}$

Solution:

$\mathop \smallint \nolimits \frac{{3{\rm{x}} + 4}}{{\sqrt {{\rm{x}} + 1} }}.{\rm{dx}}$ = $\mathop \smallint \nolimits \frac{{\left( {3{\rm{x}} + 3} \right) + 1}}{{\sqrt {{\rm{x}} + 1} }}.{\rm{dx}}$

=$\mathop \smallint \nolimits \left\{ {\frac{{3\left( {{\rm{x}} + 1} \right)}}{{{{\left( {{\rm{x}} + 1} \right)}^{\frac{1}{2}}}}} + \frac{1}{{{{\left( {{\rm{x}} + 1} \right)}^{\frac{1}{2}}}}}} \right\}.{\rm{dx}}$

= $\mathop \smallint \nolimits \left\{ {3{{\left( {{\rm{x}} + 1} \right)}^{\frac{1}{2}}} + {{\left( {{\rm{x}} + 1} \right)}^{ - \frac{1}{2}}}} \right\}.{\rm{dx}}$

=   3.$\mathop \smallint \nolimits {\left( {{\rm{x}} + 1} \right)^{\frac{1}{2}}}.{\rm{dx}} + $$\mathop \smallint \nolimits {\left( {{\rm{x}} + 1} \right)^{ - \frac{1}{2}}}$.dx

= $\frac{{3{{\left( {{\rm{x}} + 1} \right)}^{\frac{3}{2}}}}}{{\frac{3}{2}}} + \frac{{{{\left( {{\rm{x}} + 2} \right)}^{ - \frac{1}{2}}}}}{{\frac{1}{2}}}$ + c

= 2 (x + 1)^3/2 + 2(x + 1)^1/2 + c.

  

(x)$\mathop \smallint \nolimits \frac{{2{\rm{x}} + 1}}{{\sqrt {3{\rm{x}} + 2} }}.{\rm{dx}}$ 

Solution:

$\mathop \smallint \nolimits \frac{{2{\rm{x}} + 1}}{{\sqrt {3{\rm{x}} + 2} }}.{\rm{dx}}$ = $\frac{1}{3}\mathop \smallint \nolimits \frac{{6{\rm{x}} + 3}}{{\sqrt {3{\rm{x}} + 2} }}.{\rm{dx}}$

=${\rm{\: }}\frac{1}{3}\mathop \smallint \nolimits \left\{ {\frac{{2\left( {3{\rm{x}} + 2} \right)}}{{{{\left( {3{\rm{x}} + 2} \right)}^{\frac{1}{2}}}}} - \frac{1}{{{{\left( {3{\rm{x}} + 2} \right)}^{\frac{1}{2}}}}}} \right\}.{\rm{dx}}$

= $\frac{1}{2}\mathop \smallint \nolimits \left\{ {2{{\left( {3{\rm{x}} + 2} \right)}^{\frac{1}{2}}} - {{\left( {3{\rm{x}} + 2} \right)}^{ - \frac{1}{2}}}} \right\}.{\rm{dx}}$

= $\frac{2}{3}\mathop \smallint \nolimits {\left( {3{\rm{x}} + 2} \right)^{\frac{1}{2}}}.{\rm{dx}} - $$\frac{1}{3}\mathop \smallint \nolimits {\left( {3{\rm{x}} + 2} \right)^{ - \frac{1}{2}}}$.dx

= $\frac{2}{3}.\frac{{{{\left( {3{\rm{x}} + 2} \right)}^{\frac{3}{2}}}}}{{\frac{3}{2}.3}} - \frac{1}{3}.\frac{{{{\left( {3{\rm{x}} + 2} \right)}^{ - \frac{1}{2} + 1}}}}{{\frac{1}{2}.3}}$ + c

= $\frac{4}{{27}}$ (3x + 2)^3/2$--$$\frac{2}{9}$(3x + 2)^1/2 + c.

= $\frac{2}{9}\left[ {\frac{2}{3}{{\left( {3{\rm{x}} + 2} \right)}^{\frac{3}{2}}} - {{\left( {3{\rm{x}} + 2} \right)}^{\frac{1}{2}}}} \right]$ + c.

 

(xi)$\mathop \smallint \nolimits \frac{{2{\rm{x}} + 3}}{{{{\left( {3{\rm{x}} + 1} \right)}^{\frac{3}{2}}}}}.{\rm{dx}}$

Solution:

$\mathop \smallint \nolimits \frac{{2{\rm{x}} + 3}}{{{{\left( {3{\rm{x}} + 1} \right)}^{\frac{3}{2}}}}}.{\rm{dx}}$ = $\frac{1}{3}\mathop \smallint \nolimits \frac{{6{\rm{x}} + 9}}{{{{\left( {3{\rm{x}} + 1} \right)}^{\frac{3}{2}}}}}.{\rm{dx}}$

=${\rm{\: }}\frac{1}{3}\mathop \smallint \nolimits \left\{ {\frac{{2\left( {3{\rm{x}} + 1} \right)}}{{{{\left( {3{\rm{x}} + 2} \right)}^{\frac{3}{2}}}}} + \frac{7}{{{{\left( {3{\rm{x}} + 1} \right)}^{\frac{3}{2}}}}}} \right\}.{\rm{dx}}$

= $\frac{1}{2}\mathop \smallint \nolimits \left\{ {2{{\left( {3{\rm{x}} + 1} \right)}^{ - \frac{1}{2}}} + 7{{\left( {3{\rm{x}} + 1} \right)}^{ - \frac{3}{2}}}} \right\}.{\rm{dx}}$

= $\frac{2}{3}\mathop \smallint \nolimits {\left( {3{\rm{x}} + 1} \right)^{ - \frac{1}{2}}}.{\rm{dx}} + $$\frac{7}{3}$.$\mathop \smallint \nolimits {\left( {3{\rm{x}} + 1} \right)^{ - \frac{3}{2}}}$.dx

= $\frac{2}{3}.\frac{{{{\left( {3{\rm{x}} + 1} \right)}^{ - \frac{1}{2} + 1}}}}{{\frac{1}{2}.3}}.{\rm{dx}} + \frac{7}{3}.\frac{{{{\left( {3{\rm{x}} + 1} \right)}^{ - \frac{3}{2} + 1}}}}{{\frac{{ - 1}}{2}.3}}$ + c

= $\frac{4}{9}$ (3x + 1)^1/2$--$$\frac{{14}}{9}$(3x + 1)^–1/2 + c.

= $\frac{2}{9}[{\rm{\{ }}2{\left( {3{\rm{x}} + 1} \right)^{\frac{1}{2}}} - 7{\left( {3{\rm{x}} + 1} \right)^{ - \frac{1}{2}}}{\rm{]}}$ + c.

 

(xii)$\mathop \smallint \nolimits \frac{{3{\rm{x}} + 2}}{{\sqrt {5{\rm{x}} + 3} }}.{\rm{dx}}$

Solution:

$\mathop \smallint \nolimits \frac{{3{\rm{x}} + 2}}{{\sqrt {5{\rm{x}} + 3} }}.{\rm{dx}}$ = $\frac{1}{5}\mathop \smallint \nolimits \frac{{15{\rm{x}} + 10}}{{\sqrt {5{\rm{x}} + 3} }}.{\rm{dx}}$ = $\frac{1}{5}\mathop \smallint \nolimits \frac{{\left( {15{\rm{x}} + 9} \right) + 1}}{{{{\left( {5{\rm{x}} + 3} \right)}^{\frac{1}{2}}}}}.{\rm{dx}}$

=${\rm{\: }}\frac{1}{5}\mathop \smallint \nolimits \left\{ {\frac{{3\left( {5{\rm{x}} + 3} \right)}}{{{{\left( {5{\rm{x}} + 3} \right)}^{\frac{1}{2}}}}} + \frac{1}{{{{\left( {5{\rm{x}} + 3} \right)}^{\frac{1}{2}}}}}} \right\}.{\rm{dx}}$

= $\frac{1}{5}\mathop \smallint \nolimits \left\{ {3{{\left( {5{\rm{x}} + 3} \right)}^{\frac{1}{2}}} + {{\left( {5{\rm{x}} + 3} \right)}^{ - \frac{1}{2}}}} \right\}.{\rm{dx}}$

= $\frac{3}{5}\mathop \smallint \nolimits {\left( {5{\rm{x}} + 3} \right)^{\frac{1}{2}}}.{\rm{dx}} + $$\frac{1}{5}$.$\mathop \smallint \nolimits {\left( {5{\rm{x}} + 3} \right)^{ - \frac{1}{2}}}$.dx

= $\frac{3}{5}.\frac{{{{\left( {5{\rm{x}} + 3} \right)}^{\frac{3}{2}}}}}{{\frac{3}{2}.5}}.{\rm{dx}} + \frac{1}{5}.\frac{{{{\left( {5{\rm{x}} + 3} \right)}^{\frac{1}{2}}}}}{{\frac{1}{2}.5}}$ + c

= $\frac{2}{{25}}$ (5x + 3)^3/2 + $\frac{2}{{25}}$ (5x + 3)^1/2 + c

= $\frac{2}{{25}}$ [(5x + 3)^3/2 + (5x + 3)^1/2] + c.

 

5.(i)$\mathop \smallint \nolimits \frac{1}{{\sqrt {{\rm{x}} + 1}  - \sqrt {\rm{x}} }}$.dx 

Solution:

Or, $\mathop \smallint \nolimits \frac{1}{{\sqrt {{\rm{x}} + 1}  - \sqrt {\rm{x}} }}$.dx = $\mathop \smallint \nolimits \frac{1}{{\sqrt {{\rm{x}} + 1}  - \sqrt {\rm{x}} }}{\rm{*}}\frac{{\sqrt {{\rm{x}} + 1}  + \sqrt {\rm{x}} }}{{\sqrt {{\rm{x}} + 1}  + \sqrt {\rm{x}} }}.{\rm{dx}}$

= $\mathop \smallint \nolimits \frac{{{{\left( {{\rm{x}} + 1} \right)}^{\frac{1}{2}}} + {{\rm{x}}^{\frac{1}{2}}}}}{{{\rm{x}} + 1 - {\rm{x}}}}.{\rm{dx}}$ = $\mathop \smallint \nolimits {\left( {{\rm{x}} + 1} \right)^{\frac{1}{2}}}.{\rm{dx}} + \mathop \smallint \nolimits {{\rm{x}}^{\frac{1}{2}}}.{\rm{dx}}$

= $\frac{{{{\left( {{\rm{x}} + 1} \right)}^{\frac{3}{2}}}}}{{\frac{3}{2}}} + \frac{{{{\rm{x}}^{\frac{3}{2}}}}}{{\frac{3}{2}}}$ + c = $\frac{2}{3}$[(x + 1)^3/2 + x^3/2] + c.

 

(ii) $\mathop \smallint \nolimits \frac{{{\rm{dx}}}}{{\sqrt {{\rm{x}} + {\rm{a}}}  - \sqrt {{\rm{x}} - {\rm{a}}} }}$.

Solution:

Or, $\mathop \smallint \nolimits \frac{{{\rm{dx}}}}{{\sqrt {{\rm{x}} + {\rm{a}}}  - \sqrt {{\rm{x}} - {\rm{a}}} }}$.= $\mathop \smallint \nolimits \frac{{{\rm{dx}}}}{{\sqrt {{\rm{x}} + {\rm{a}}}  - \sqrt {{\rm{x}} - {\rm{a}}} }}{\rm{*}}\frac{{\sqrt {{\rm{x}} + {\rm{a}}}  + \sqrt {{\rm{x}} - {\rm{a}}} }}{{\sqrt {{\rm{x}} + {\rm{a}}}  + \sqrt {{\rm{x}} - {\rm{a}}} }}$

= $\mathop \smallint \nolimits \frac{{\sqrt {{\rm{x}} + 1}  + \sqrt {{\rm{x}} - {\rm{a}}} }}{{{\rm{x}} + {\rm{a}} - {\rm{x}} + {\rm{a}}}}.{\rm{dx}}$

= $\mathop \smallint \nolimits \left[ {{{\left( {{\rm{x}} + {\rm{a}}} \right)}^{\frac{1}{2}}} + {{\left( {{\rm{x}} - {\rm{a}}} \right)}^{\frac{1}{2}}}} \right].{\rm{dx}}$

= \[\]

= $\frac{1}{{2{\rm{a}}}}$.$\left[ {\frac{{{{\left( {{\rm{x}} + {\rm{a}}} \right)}^{\frac{3}{2}}}}}{{\frac{3}{2}}} + \frac{{{{\left( {{\rm{x}} - {\rm{a}}} \right)}^{\frac{3}{2}}}}}{{\frac{3}{2}}}} \right]$ + c.

= $\frac{1}{{2{\rm{a}}}}$.$\frac{2}{3}$ [(x + a)^3/2 + (x – a)^3/2] + c.

=$\frac{1}{{3{\rm{a}}}}$[(x + a)^3/2 + (x – a)^3/2] + c.

 

(iii)$\mathop \smallint \nolimits \frac{1}{{\sqrt {2{\rm{x}} + 1}  - \sqrt {2{\rm{x}} - 3} }}$.dx

Solution:

Or, $\mathop \smallint \nolimits \frac{1}{{\sqrt {2{\rm{x}} + 1}  - \sqrt {2{\rm{x}} - 3} }}$.dx = $\mathop \smallint \nolimits \frac{1}{{\sqrt {2{\rm{x}} + 1}  - \sqrt {2{\rm{x}} - 3} }}{\rm{*}}\frac{{\sqrt {2{\rm{x}} + 1}  + \sqrt {2{\rm{x}} - 3} }}{{\sqrt {2{\rm{x}} + 1}  + \sqrt {2{\rm{x}} - 3} }}.{\rm{dx}}$

= $\mathop \smallint \nolimits \frac{{\sqrt {2{\rm{x}} + 1}  + \sqrt {2{\rm{x}} - 3} }}{{2{\rm{x}} + 1 - 2{\rm{x}} + 3}}.{\rm{dx}}$

= $\frac{1}{4}.[\mathop \smallint \nolimits {\left( {2{\rm{x}} + 1} \right)^{\frac{1}{2}}}.{\rm{dx}} + \mathop \smallint \nolimits {\left( {2{\rm{x}} - 3} \right)^{\frac{1}{2}}}].{\rm{dx}}$

= $\frac{1}{4}$.$\left[ {\frac{{{{\left( {2{\rm{x}} + 1} \right)}^{\frac{3}{2}}}}}{{\frac{3}{2}.2}} + \frac{{{{\left( {2{\rm{x}} - 3} \right)}^{\frac{3}{2}}}}}{{\frac{3}{2}.2}}} \right]$ + c.

= $\frac{1}{{12}}$ [(2x + 1)^3/2 + (2x – 3)^3/2] + c.

 

(iv)$\mathop \smallint \nolimits \frac{{{\rm{dx}}}}{{\sqrt {{\rm{x}} + {\rm{a}}}  - \sqrt {{\rm{x}} - {\rm{b}}} }}$

Solution:

Or, $\mathop \smallint \nolimits \frac{{{\rm{dx}}}}{{\sqrt {{\rm{x}} + {\rm{a}}}  - \sqrt {{\rm{x}} - {\rm{b}}} }}$ = $\mathop \smallint \nolimits \frac{{{\rm{dx}}.}}{{\sqrt {{\rm{x}} + {\rm{a}}}  - \sqrt {{\rm{x}} - {\rm{b}}} }}{\rm{*}}\frac{{\sqrt {{\rm{x}} + {\rm{a}}}  + \sqrt {{\rm{x}} - {\rm{b}}} }}{{\sqrt {{\rm{x}} + {\rm{a}}}  + \sqrt {{\rm{x}} - {\rm{b}}} }}.{\rm{dx}}$

= $\mathop \smallint \nolimits \frac{{\sqrt {{\rm{x}} + {\rm{a}}}  + \sqrt {{\rm{x}} - {\rm{b}}} }}{{{\rm{x}} + {\rm{a}} - {\rm{x}} + {\rm{b}}}}.{\rm{dx}}$

= $\frac{1}{{{\rm{a}} + {\rm{b}}}}.\left[ {\mathop \smallint \nolimits {{\left( {{\rm{x}} + {\rm{a}}} \right)}^{\frac{1}{2}}}.{\rm{dx}} + \mathop \smallint \nolimits {{\left( {{\rm{x}} - {\rm{b}}} \right)}^{\frac{1}{2}}}.{\rm{dx}}} \right]$

= $\frac{1}{{{\rm{a}} + {\rm{b}}}}$.$\left[ {\frac{{{{\left( {{\rm{x}} + {\rm{a}}} \right)}^{\frac{3}{2}}}}}{{\frac{3}{2}}} + \frac{{{{\left( {{\rm{x}} - {\rm{b}}} \right)}^{\frac{3}{2}}}}}{{\frac{3}{2}}}} \right]$ + c.

= $\frac{1}{{{\rm{a}} + {\rm{b}}}}$$\left[ {\frac{2}{3}{{\left( {{\rm{x}} + {\rm{a}}} \right)}^{\frac{3}{2}}} + \frac{2}{3}{{\left( {{\rm{x}} - {\rm{b}}} \right)}^{\frac{3}{2}}}} \right]$ + c

= $\frac{2}{{3\left( {{\rm{a}} + {\rm{b}}} \right)}}$$\left[ {{{\left( {{\rm{x}} + {\rm{a}}} \right)}^{\frac{3}{2}}} + {{\left( {{\rm{x}} - {\rm{b}}} \right)}^{\frac{3}{2}}}} \right]$ + c.

 

 

6.(i)$\mathop \smallint \nolimits {\rm{sin}}5{\rm{x}}$ dx

Solution:

Or, $\mathop \smallint \nolimits {\rm{sin}}5{\rm{x}}$ dx = $ - \frac{{{\rm{cos}}5{\rm{x}}}}{5}$ + c.

                                    

(ii)$\mathop \smallint \nolimits {\rm{sin}}\left( {{\rm{ax}} + {\rm{b}}} \right).{\rm{dx}}$

Solution:

Or, $\mathop \smallint \nolimits {\rm{sin}}\left( {{\rm{ax}} + {\rm{b}}} \right).{\rm{dx}}$ = $ - \frac{{{\rm{cos}}\left( {{\rm{ax}} + {\rm{b}}} \right)}}{{\rm{a}}}$ + c.

 

(iii)$\mathop \smallint \nolimits {\rm{sin}}\left( {{{\rm{a}}^2}{\rm{x}} + {\rm{b}}} \right).{\rm{dx}}$

Solution:

Or, $\mathop \smallint \nolimits {\rm{sin}}\left( {{{\rm{a}}^2}{\rm{x}} + {\rm{b}}} \right).{\rm{dx}}$ = $\frac{{{\rm{sin}}\left( {{{\rm{a}}^2}{\rm{x}} + {\rm{b}}} \right)}}{{{{\rm{a}}^2}}}$ + c.

 

(iv)$\mathop \smallint \nolimits {\sec ^2}\left( {2{\rm{x}} + 3} \right).{\rm{dx}}$

Solution:

Or, $\mathop \smallint \nolimits {\sec ^2}\left( {2{\rm{x}} + 3} \right).{\rm{dx}}$ = $\frac{{{\rm{tan}}\left( {2{\rm{x}} + 3} \right)}}{2}$ + c.

 

(v)$\mathop \smallint \nolimits {\sin ^2}{\rm{ax}}.{\rm{dx}}$ 

Solution:

Or, $\mathop \smallint \nolimits {\sin ^2}{\rm{ax}}.{\rm{dx}}$ = $\mathop \smallint \nolimits \frac{{1 - {\rm{cos}}2{\rm{ax}}}}{2}.{\rm{dx}}$ = $\frac{1}{2}\mathop \smallint \nolimits \left( {1 - {\rm{cos}}2{\rm{ax}}} \right){\rm{dx}}$

= $\frac{1}{2}[\mathop \smallint \nolimits 1.{\rm{dx}} - \mathop \smallint \nolimits {\rm{cos}}2{\rm{ax}}.{\rm{dx}}]{\rm{\: }}$ = $\frac{1}{2}\left[ {{\rm{x}} - \frac{{{\rm{sin}}2{\rm{ax}}}}{{2{\rm{a}}}}} \right]$ + c.

 

(vi)$\mathop \smallint \nolimits {\cos ^3}{\rm{bx}}.{\rm{dx}}$ 

Solution:

Or, $\mathop \smallint \nolimits {\cos ^3}{\rm{bx}}.{\rm{dx}}$ = $\mathop \smallint \nolimits \frac{{1 + {\rm{cos}}2{\rm{bx}}}}{2}.{\rm{dx}}$

= $\frac{1}{2}[\mathop \smallint \nolimits 1.{\rm{dx}} + \mathop \smallint \nolimits {\rm{cos}}2{\rm{bx}}.{\rm{dx}}]{\rm{\: }}$ = $\frac{1}{2}\left[ {{\rm{x}} - \frac{{{\rm{sin}}2{\rm{bx}}}}{{2{\rm{b}}}}} \right]$ + c.

 

(vii)$\mathop \smallint \nolimits {\tan ^2}{\rm{ax}}.{\rm{dx}}$ 

Solution:

Or, $\mathop \smallint \nolimits {\tan ^2}{\rm{ax}}.{\rm{dx}}$ = $\mathop \smallint \nolimits ({\sec ^2}{\rm{ax}}.{\rm{dx}} - \mathop \smallint \nolimits 1.{\rm{dx}}$  = $\frac{{{\rm{tanax}}}}{{\rm{a}}}$ – x + c.

 

(viii)$\mathop \smallint \nolimits {\sin ^4}.{\rm{dx}}$ 

Solution:

Or, $\mathop \smallint \nolimits {\sin ^4}.{\rm{dx}}$ = $\frac{1}{4}\mathop \smallint \nolimits 4{\sin ^4}{\rm{x}}.{\rm{dx}}$ = $\frac{1}{4}\mathop \smallint \nolimits {\left( {2{{\sin }^2}{\rm{x}}} \right)^2}{\rm{dx}}$

= $\frac{1}{4}[\mathop \smallint \nolimits {\left( {1 - {\rm{cos}}2{\rm{x}}} \right)^2}.{\rm{dx}}$

= $\frac{1}{4}\mathop \smallint \nolimits (1 - 2{\rm{cos}}2{\rm{x}} + {\cos ^2}2{\rm{x}}).$dx

= $\frac{1}{4}\mathop \smallint \nolimits [1 - 2{\rm{cos}}2{\rm{x}} + \frac{1}{2}(2{\cos ^2}2{\rm{x}})]$.dx

= $\frac{1}{4}\mathop \smallint \nolimits \left[ {1 - 2{\rm{cos}}2{\rm{x}} + \frac{1}{2}\left( {1 + {\rm{cos}}4{\rm{x}}} \right)} \right]$.dx

= $\frac{1}{4}$$\mathop \smallint \nolimits \left( {1 - 2{\rm{cos}}2{\rm{x}} + \frac{1}{2} + \frac{1}{2}{\rm{cos}}4{\rm{x}}} \right)$.dx

= $\frac{1}{4}\mathop \smallint \nolimits \left( {\frac{3}{2} - 2.{\rm{cos}}2{\rm{x}} + \frac{1}{2}.{\rm{cos}}4{\rm{x}}} \right)$.dx

= $\frac{1}{4}$$\left[ {\frac{{3{\rm{x}}}}{2} - \frac{{2{\rm{sin}}2{\rm{x}}}}{2} + \frac{1}{2}.\frac{{{\rm{sin}}4{\rm{x}}}}{4}} \right]$ + c

= $\frac{1}{4}\left[ {\frac{{12{\rm{x}} - 8{\rm{sin}}2{\rm{x}} + {\rm{sin}}4{\rm{x}}}}{8}} \right]$ + c

= $\frac{1}{{32}}$ (12x – 8sin2x + sin4x) + c.

 

(ix) $\mathop \smallint \nolimits {\cos ^4}{\rm{nx}}.{\rm{dx}}$

Solution:

Or, $\mathop \smallint \nolimits {\cos ^4}{\rm{nx}}.{\rm{dx}}$ = $\frac{1}{4}\mathop \smallint \nolimits 4{\cos ^4}{\rm{nxdx}}$ = $\frac{1}{4}\mathop \smallint \nolimits {\left( {2{{\cos }^2}{\rm{nx}}} \right)^2}{\rm{dx}}$

= $\frac{1}{4}[\mathop \smallint \nolimits {\left( {1 + {\rm{cos}}2{\rm{nx}}} \right)^2}.{\rm{dx}}$

= $\frac{1}{4}\mathop \smallint \nolimits (1 + 2{\rm{cos}}2{\rm{nx}} + \frac{1}{2}({\cos ^2}2{\rm{nx}}).$dx

= $\frac{1}{4}\mathop \smallint \nolimits (1 + 2{\rm{cos}}2{\rm{nx}} + \frac{1}{2}(2{\cos ^2}2{\rm{nx}})){\rm{dx}}$

= $\frac{1}{4}\mathop \smallint \nolimits \left[ {1 + 2{\rm{cos}}2{\rm{nx}} + \frac{1}{2}\left( {1 + {\rm{cos}}4{\rm{nx}}} \right)} \right]$.dx

= $\frac{1}{4}\mathop \smallint \nolimits \left( {\frac{3}{2} + 2.{\rm{cos}}2{\rm{nx}} + \frac{1}{2}.{\rm{cos}}4{\rm{nx}}} \right)$.dx

= $\frac{1}{4}$$\left[ {\frac{{3{\rm{x}}}}{2} + \frac{{2{\rm{sin}}2{\rm{nx}}}}{{2{\rm{n}}}} + \frac{1}{2}.\frac{{{\rm{sin}}4{\rm{nx}}}}{{4{\rm{n}}}}} \right]$ + c

= $\frac{1}{4}\left[ {\frac{{12{\rm{x}} + 8{\rm{sin}}2{\rm{nx}} + {\rm{sin}}4{\rm{nx}}}}{{8{\rm{n}}}}} \right]$ + c

= $\frac{1}{{32{\rm{n}}}}$ (12nx + 8sin2nx + sin4nx) + c.

 

(x)$\frac{1}{{{{\cos }^2}{\rm{x}}.{{\sin }^2}{\rm{x}}}}$.dx 

Solution:

Or, $\frac{1}{{{{\cos }^2}{\rm{x}}.{{\sin }^2}{\rm{x}}}}$.dx = $\mathop \smallint \nolimits \frac{{{{\sin }^2}{\rm{x}} + {{\cos }^2}{\rm{x}}}}{{{{\cos }^2}{\rm{x}}.{{\sin }^2}{\rm{x}}}}$.dx  (sin²x + cos²x = 1)

= $\frac{1}{4}\mathop \smallint \nolimits \{ \frac{{{{\sin }^2}{\rm{x}}}}{{{{\cos }^2}{\rm{x}}.{{\sin }^2}{\rm{x}}}} + \frac{{{{\cos }^2}{\rm{x}}}}{{{{\cos }^2}.{{\sin }^2}{\rm{x}}}}\} .{\rm{dx}}$

= $\mathop \smallint \nolimits ({\sec ^2}{\rm{x}} + {\rm{cose}}{{\rm{c}}^2}{\rm{x}}).$dx

= $\mathop \smallint \nolimits {\sec ^2}{\rm{x}}.{\rm{dx}}$ + $\mathop \smallint \nolimits {\rm{cose}}{{\rm{c}}^2}{\rm{x}}$.dx = tanx + (–cotx) + c

= tanx – cotx + c.

  

(xi)$\frac{1}{{{{\sec }^2}{\rm{x}}.{{\tan }^2}{\rm{x}}}}$.dx 

Solution:

Or, $\frac{1}{{{{\sec }^2}{\rm{x}}.{{\tan }^2}{\rm{x}}}}$.dx = $\mathop \smallint \nolimits \frac{{{{\sec }^2}{\rm{x}} - {{\tan }^2}{\rm{x}}}}{{{{\sec }^2}{\rm{x}}.{{\tan }^2}{\rm{x}}}}$.dx  (sec²x – tan²x = 1)

= $\mathop \smallint \nolimits (\frac{{{{\sec }^2}{\rm{x}}}}{{{{\sec }^2}{\rm{x}}.{{\tan }^2}{\rm{x}}}} - \frac{{{{\tan }^2}{\rm{x}}}}{{{{\sec }^2}.{{\tan }^2}{\rm{x}}}}).{\rm{dx}}$

= $\mathop \smallint \nolimits ({\cot ^2}{\rm{x}} - {\cos ^2}{\rm{x}}).$dx

= $\mathop \smallint \nolimits [{\rm{cose}}{{\rm{c}}^2}{\rm{x}} - 1 - \frac{1}{2}\left( {2{{\cos }^2}{\rm{x}}} \right)$.dx

= $\mathop \smallint \nolimits {\rm{cose}}{{\rm{c}}^2}{\rm{x}} - 1 - \frac{1}{2}\left( {1 + {\rm{cos}}2{\rm{x}}} \right)$.dx

= $\mathop \smallint \nolimits ({\rm{cose}}{{\rm{c}}^2}{\rm{x}} - \frac{3}{2} - \frac{1}{2}{\rm{cos}}2{\rm{x}})$.dx

= –cotx – $\frac{3}{2}$x – $\frac{1}{2}$.$\frac{{{\rm{sin}}2{\rm{x}}}}{2}$ + c

= –cotx – $\frac{{3{\rm{x}}}}{2}$ – $\frac{1}{4}$sin2x + c.

 

7(i)$\mathop \smallint \nolimits \sqrt {1 + {\rm{cosnx}}} $.dx 

Solution:

Or, $\mathop \smallint \nolimits \sqrt {1 + {\rm{cosnx}}} $.dx = $\mathop \smallint \nolimits \sqrt {2{{\cos }^2}\frac{{{\rm{nx}}}}{2}} $ = $\sqrt 2 $$\mathop\smallint\nolimits^\cos\frac{{{\rm{nx}}}}{2}$.dx

= $\sqrt 2 $$\frac{{\sin \frac{{{\rm{nx}}}}{2}}}{{\frac{{\rm{n}}}{2}}}$ + c = $\frac{{2\sqrt 2}}{{\rm{n}}}$.sin$\frac{{{\rm{nx}}}}{2}$ + c

 

(ii)$\mathop \smallint \nolimits \sqrt {1 - {\rm{cospx}}} $.dx

Solution:

Or, $\mathop \smallint \nolimits \sqrt {1 - {\rm{cospx}}} $.dx = $\mathop \smallint \nolimits \sqrt {2{{\sin }^2}\frac{{{\rm{px}}}}{2}} $.dx

= $\sqrt 2 $$\left( {\frac{{ - \cos \frac{{{\rm{px}}}}{2}}}{{\frac{{\rm{p}}}{2}}}} \right)$ + c = $ - \frac{{2\sqrt 2 }}{{\rm{p}}}$.cos $\frac{{{\rm{px}}}}{2}$ + c

 

(iii)$\mathop \smallint \nolimits \sqrt {1 + {\rm{sin}}2{\rm{ax}}} $.dx

Solution:

Or, $\mathop \smallint \nolimits \sqrt {1 + {\rm{sin}}2{\rm{ax}}} $.dx = $\mathop \smallint \nolimits \sqrt {({{\sin }^2}{\rm{ax}} + {{\cos }^2}{\rm{ax}} + 2{\rm{sinax}}.{\rm{coax}}} $.dx

= $\sqrt {{{\left( {{\rm{sinax}} + {\rm{cosax}}} \right)}^2}} $.dx

=  $\mathop \smallint \nolimits \left( {{\rm{sinax}} + {\rm{cosax}}} \right)$.dx

= $ - \frac{{{\rm{cosax}}}}{{\rm{a}}}$ + $\frac{{{\rm{sinax}}}}{{\rm{a}}}$ + c = $\frac{1}{{\rm{a}}}$(sinax – cosax) + c

 

(iv)$\mathop \smallint \nolimits \frac{{{\rm{dx}}}}{{1 + {\rm{cosmx}}}}$

Solution:

Or, $\mathop \smallint \nolimits \frac{{{\rm{dx}}}}{{1 + {\rm{cosmx}}}}$ = $\mathop \smallint \nolimits \frac{{{\rm{dx}}}}{{2{\rm{co}}{{\rm{s}}^2}\frac{{{\rm{mx}}}}{2}}}$ = $\frac{1}{2}\mathop \smallint \nolimits {\sec ^2}\frac{{{\rm{mx}}}}{2}.{\rm{dx\: \: }}$

= $\frac{1}{2}\left( {\frac{{\tan \frac{{{\rm{mx}}}}{2}}}{{\frac{{\rm{m}}}{2}}}} \right)$ + c.

= $\frac{1}{{\rm{m}}}$tan $\frac{{{\rm{mx}}}}{2}$ + c.

 

(v)$\mathop \smallint \nolimits \frac{{{\rm{dx}}}}{{1 - {\rm{cosnx}}}}$

Solution:

Or, $\mathop \smallint \nolimits \frac{{{\rm{dx}}}}{{1 - {\rm{cosnx}}}}$ = $\mathop \smallint \nolimits \frac{{{\rm{dx}}}}{{2{{\sin }^2}\frac{{{\rm{nx}}}}{2}}}$ = $\frac{1}{2}\mathop \smallint \nolimits {\rm{cose}}{{\rm{c}}^{2{\rm{\: }}}}\frac{{{\rm{nx}}}}{2}.{\rm{dx\: \: }}$

= $\frac{1}{2}\left( {\frac{{ - \cot \frac{{{\rm{nx}}}}{2}}}{{\frac{{\rm{n}}}{2}}}} \right)$ + c.

= $ - \frac{1}{{\rm{n}}}$cot. $\frac{{{\rm{nx}}}}{2}$ + c

 

(vi)$\mathop \smallint \nolimits \frac{{{\rm{dx}}}}{{1 - {\rm{sinax}}}}$ 

Solution:

Or, $\mathop \smallint \nolimits \frac{{{\rm{dx}}}}{{1 - {\rm{sinax}}}}$ = $\mathop \smallint \nolimits \frac{{{\rm{dx}}}}{{1 - {\rm{sinax}}}}{\rm{*}}\frac{{1 + {\rm{sinax}}}}{{1 + {\rm{sinax}}}}$

 = $\mathop \smallint \nolimits \frac{{1 + {\rm{sinax}}}}{{1 - {{\sin }^2}{\rm{ax}}}}.{\rm{dx\: \: }}$

= $\mathop \smallint \nolimits \frac{{1 + {\rm{sinax}}}}{{{{\cos }^2}{\rm{ax}}}}.{\rm{dx\: \: }}$

= $\mathop \smallint \nolimits \left( {\frac{1}{{{{\cos }^2}{\rm{ax}}}} + \frac{{{\rm{sinax}}}}{{{{\cos }^2}{\rm{ax}}}}} \right).{\rm{dx\: \: }}$

= $\mathop \smallint \nolimits \left( {{{\sec }^2}{\rm{ax}} + {\rm{tanax}}.{\rm{secax}}} \right).{\rm{dx\: \: }}$

= $\frac{{{\rm{tanax}}}}{{\rm{a}}} + \frac{{{\rm{secax}}}}{{\rm{a}}}$ + c.

= $\frac{1}{{\rm{a}}}$(tanax + secax) + c.

 

8(i)$\mathop \smallint \nolimits {\rm{sin}}6{\rm{x}}.{\rm{cos}}2{\rm{x}}.{\rm{dx}}$ 

Solution:

Or, $\mathop \smallint \nolimits {\rm{sin}}6{\rm{x}}.{\rm{cos}}2{\rm{x}}.{\rm{dx}}$ = $\frac{1}{2}\mathop \smallint \nolimits 2.{\rm{sin}}6{\rm{x}}.{\rm{cos}}2{\rm{x}}.{\rm{dx}}$

= $\frac{1}{2}\mathop \smallint \nolimits \left[ {\sin \left( {6{\rm{x}} + 2{\rm{x}}} \right) + \sin \left( {6{\rm{x}} - 2{\rm{x}}} \right)} \right].{\rm{dx}}$

= $\frac{1}{2}\mathop \smallint \nolimits \left( {{\rm{sin}}8{\rm{x}} + {\rm{sin}}4{\rm{x}}} \right).{\rm{dx}}$

= $\frac{1}{2}\left[ { - \frac{{{\rm{cos}}8{\rm{x}}}}{8} - \frac{{{\rm{cos}}4{\rm{x}}}}{4}} \right]$ + c

= $\frac{{ - {\rm{cos}}8{\rm{x}}}}{{16}} - \frac{{{\rm{cos}}4{\rm{x}}}}{8}$ + c.

 

(ii)$\mathop \smallint \nolimits {\rm{sin}}7{\rm{x}}.{\rm{cos}}5{\rm{x}}.{\rm{dx}}$ 

Solution:

Or, $\mathop \smallint \nolimits {\rm{sin}}7{\rm{x}}.{\rm{cos}}5{\rm{x}}.{\rm{dx}}$ = $\frac{1}{2}\mathop \smallint \nolimits 2.{\rm{sin}}7{\rm{x}}.{\rm{sin}}5{\rm{x}}.{\rm{dx}}$

= $\frac{1}{2}\mathop \smallint \nolimits \left[ {\sin \left( {7{\rm{x}} - 5{\rm{x}}} \right) - \cos \left( {7{\rm{x}} + 5{\rm{x}}} \right)} \right].{\rm{dx}}$

= $\frac{1}{2}\mathop \smallint \nolimits \left( {{\rm{cos}}2{\rm{x}} - {\rm{cos}}12{\rm{x}}} \right).{\rm{dx}}$

= $\frac{1}{2}\left[ {\frac{{{\rm{sin}}2{\rm{x}}}}{2} - \frac{{{\rm{sin}}12{\rm{x}}}}{{12}}} \right]$ + c

= $\frac{{{\rm{sin}}2{\rm{x}}}}{4} - \frac{{{\rm{sin}}12{\rm{x}}}}{{24}}$ + c.

 

(iii)$\mathop \smallint \nolimits {\rm{sin}}6{\rm{x}}.{\rm{cos}}8{\rm{x}}.{\rm{dx}}$ 

Solution:

Or, $\mathop \smallint \nolimits {\rm{sin}}6{\rm{x}}.{\rm{cos}}8{\rm{x}}.{\rm{dx}}$ = $\frac{1}{2}\mathop \smallint \nolimits 2.{\rm{sin}}6{\rm{x}}.{\rm{cos}}8{\rm{x}}.{\rm{dx}}$

= $\frac{1}{2}\mathop \smallint \nolimits \left[ {\sin \left( {6{\rm{x}} + 8{\rm{x}}} \right) + {\rm{sin}}\left( {6{\rm{x}} - 8{\rm{x}}} \right)} \right].{\rm{dx}}$

= $\frac{1}{2}\mathop \smallint \nolimits \left( {{\rm{sin}}14{\rm{x}} + {\rm{sin}}( - 2{\rm{x}}} \right).{\rm{dx}}$

= $\frac{1}{2}\mathop \smallint \nolimits \left( {{\rm{sin}}14{\rm{x}} - {\rm{sin}}2{\rm{x}}} \right).{\rm{dx}}$

= $\frac{1}{2}\left[ { - \frac{{{\rm{cos}}14{\rm{x}}}}{{14}} - \frac{{\left( { - {\rm{cos}}2{\rm{x}}} \right)}}{2}} \right]$ + c

= $ - \frac{{{\rm{cos}}14{\rm{x}}}}{{28}} + \frac{{{\rm{cos}}2{\rm{x}}}}{4}$ + c = $\frac{{{\rm{cos}}2{\rm{x}}}}{4} - \frac{{{\rm{cos}}14{\rm{x}}}}{{18}}$ + c.

 

(iv)$\mathop \smallint \nolimits {\rm{cospx}}.{\rm{cosqx}}.{\rm{dx}}$ 

Solution:

Or, $\mathop \smallint \nolimits {\rm{cospx}}.{\rm{cosqx}}.{\rm{dx}}$ = $\frac{1}{2}\mathop \smallint \nolimits 2.{\rm{cospx}}.{\rm{cosqx}}.{\rm{dx}}$

= $\frac{1}{2}\mathop \smallint \nolimits \left[ {\cos \left( {{\rm{px}} + {\rm{qx}}} \right) + {\rm{cos}}\left( {{\rm{px}} - {\rm{qx}}} \right)} \right].{\rm{dx}}$

= $\frac{1}{2}\mathop \smallint \nolimits \left( {\cos \left( {{\rm{p}} + {\rm{q}}} \right){\rm{x}} + {\rm{cos}}({\rm{p}} - {\rm{q}}} \right){\rm{x}}).{\rm{dx}}$

= $\frac{1}{2}\left[ {\frac{{{\rm{sin}}\left( {{\rm{p}} + {\rm{q}}} \right){\rm{x}}}}{{{\rm{p}} + {\rm{q}}}} + \frac{{\sin \left( {{\rm{p}} - {\rm{q}}} \right){\rm{x}}}}{{{\rm{p}} - {\rm{q}}}}} \right]$ + c

 

9.(i)$\mathop \smallint \nolimits \left( {{{\rm{e}}^{{\rm{px}}}} + {{\rm{e}}^{ - {\rm{qx}}}}} \right).{\rm{dx}}$ 

Solution:

$\mathop \smallint \nolimits \left( {{{\rm{e}}^{{\rm{px}}}} + {{\rm{e}}^{ - {\rm{qx}}}}} \right).{\rm{dx}}$ = $\mathop \smallint \nolimits {{\rm{e}}^{{\rm{px}}}}.{\rm{dx}}$ + $\mathop \smallint \nolimits {{\rm{e}}^{ - {\rm{qx}}}}$.dx

= $\frac{{{{\rm{e}}^{{\rm{px}}}}}}{{\rm{p}}} + \frac{{{{\rm{e}}^{ - {\rm{qx}}}}}}{{ - {\rm{q}}}}$ + c = $\frac{{{{\rm{e}}^{{\rm{px}}}}}}{{\rm{p}}} - \frac{{{{\rm{e}}^{ - {\rm{qx}}}}}}{{\rm{q}}}$ + c.

 

(ii)$\mathop \smallint \nolimits {\left( {{{\rm{e}}^{{\rm{px}}}} + {{\rm{e}}^{ - {\rm{px}}}}} \right)^2}$.dx

Solution:

$\mathop \smallint \nolimits {\left( {{{\rm{e}}^{{\rm{px}}}} + {{\rm{e}}^{ - {\rm{px}}}}} \right)^2}$.dx

= $\mathop \smallint \nolimits \left\{ {{{\left( {{{\rm{e}}^{{\rm{px}}}}} \right)}^2} + 2{{\rm{e}}^{{\rm{px}}}}.{{\rm{e}}^{ - {\rm{px}}}} + {{\left( {{{\rm{e}}^{ - {\rm{px}}}}} \right)}^2}} \right\}.{\rm{dx}}$

= $\mathop \smallint \nolimits \left( {{{\rm{e}}^{2{\rm{px}}}} + 2 + {{\rm{e}}^{ - 2{\rm{px}}}}} \right)$.dx

= $\frac{{{{\rm{e}}^{2{\rm{px}}}}}}{{2{\rm{p}}}}$ + 2x + $\frac{{{{\rm{e}}^{ - 2{\rm{px}}}}}}{{ - 2{\rm{p}}}}$ + c.

= $\frac{{{{\rm{e}}^{2{\rm{px}}}}}}{{2{\rm{p}}}}$ + 2x – $\frac{{{{\rm{e}}^{ - 2{\rm{px}}}}}}{{2{\rm{p}}}}$ + c

 

(iii)$\mathop \smallint \nolimits \frac{{{{\rm{e}}^{2{\rm{x}}}} + {{\rm{e}}^{\rm{x}}} + 1}}{{{{\rm{e}}^{\rm{x}}}}}$.dx

Solution:

$\mathop \smallint \nolimits \frac{{{{\rm{e}}^{2{\rm{x}}}} + {{\rm{e}}^{\rm{x}}} + 1}}{{{{\rm{e}}^{\rm{x}}}}}$.dx

= $\mathop \smallint \nolimits \left( {\frac{{{{\rm{e}}^{2{\rm{x}}}}}}{{{{\rm{e}}^{\rm{x}}}}} + \frac{{{{\rm{e}}^{\rm{x}}}}}{{{{\rm{e}}^{\rm{x}}}}} + \frac{1}{{{{\rm{e}}^{\rm{x}}}}}} \right).{\rm{dx}}$

= $\mathop \smallint \nolimits \left( {{{\rm{e}}^{\rm{x}}} + 1 + {{\rm{e}}^{ - {\rm{x}}}}} \right)$.dx

= ${{\rm{e}}^{\rm{x}}}$ + x + $\frac{{{{\rm{e}}^{ - {\rm{x}}}}}}{{ - 1}}$ + c = e^x + x – e^–x + c.

 

(iv)$\mathop \smallint \nolimits {{\rm{e}}^{3{\rm{x}}}}$.dx + $\mathop \smallint \nolimits {{\rm{e}}^{\rm{x}}}$.dx

Solution:

$\mathop \smallint \nolimits {{\rm{e}}^{3{\rm{x}}}}$.dx + $\mathop \smallint \nolimits {{\rm{e}}^{\rm{x}}}$.dx

= $\frac{{{{\rm{e}}^{3{\rm{x}}}}}}{3}$ + e^x + c