Question: $\mathop \smallint \nolimits^ \left( {2{\rm{x}} + 1} \right)\sqrt {4{{\rm{x}}^2} + 20{\rm{x}} + 21} $.dx
Solution:
I = $\mathop \smallint \nolimits^ \left( {2{\rm{x}} + 1}
\right)\sqrt {4{{\rm{x}}^2} + 20{\rm{x}} + 21} $.dx = $\frac{1}{4}\mathop
\smallint \nolimits^ \left( {8{\rm{x}} + 4} \right).\sqrt {4{{\rm{x}}^2} +
20{\rm{x}} + 21} .{\rm{dx}}$.
= $\frac{1}{4}$$\mathop \smallint \nolimits^ \left(
{8{\rm{x}} + 20} \right) - 16.\sqrt {4{{\rm{x}}^2} + 20{\rm{x}} + 21} $.dx
= $\frac{1}{4}$$\mathop \smallint \nolimits^ \left(
{8{\rm{x}} + 20} \right).\sqrt {4{{\rm{x}}^2} + 20{\rm{x}} + 21} $.dx –
4$\mathop \smallint \nolimits^ \sqrt {4{{\rm{x}}^2} + 20{\rm{x}} + 21} $.dx
= I1 – I2.
I1 = $\frac{1}{4}$$\mathop \smallint
\nolimits^ \left( {8{\rm{x}} + 20} \right).\sqrt {4{{\rm{x}}^2} + 20{\rm{x}} +
21} $.dx
Put 4x2 + 20x + 21 = y.
(8x + 20).dx = dy.
I1 = $\frac{1}{4}\mathop \smallint
\nolimits^ {{\rm{y}}^{\frac{1}{2}}}$.dy =
$\frac{1}{4}$.$\frac{{{{\rm{y}}^{\frac{3}{2}}}}}{{\frac{3}{2}}}$ + c1 =
$\frac{1}{6}$(4x2 + 20x + 21)3/2 + c1
I2 = $4\mathop \smallint \nolimits^ \sqrt
{4{{\rm{x}}^2} + 20{\rm{x}} + 21} $.dx
= 4 $\mathop \smallint \nolimits^ \sqrt {\left(
{4{{\rm{x}}^2} + 20{\rm{x}} + 25} \right) - 4} $.dx = 4$\mathop \smallint
\nolimits^ \sqrt {{{\left( {2{\rm{x}} + 5} \right)}^2} - {{\left( 2
\right)}^2}} $.dx
Put 2x + 5 = y.
So, dx = $\frac{1}{2}$.dy
I2 = 2$\mathop \smallint \nolimits^ \sqrt
{{{\left( {\rm{y}} \right)}^2} - {{\left( 2 \right)}^2}} $.dy
= 2$\left\{ {\frac{1}{2}{\rm{y}}\sqrt {{{\left( {\rm{y}}
\right)}^2} - {{\left( 2 \right)}^2}} - \frac{{{{\left( 2
\right)}^2}}}{2}.\log \left( {{\rm{y}} + \sqrt {{{\rm{y}}^2} - 4} } \right)}
\right\}$ + c2.
= y.$\sqrt {{{\rm{y}}^2} - 4} $ – 4.log (y + $\sqrt
{{{\rm{y}}^2} - 4} $) + c2.
= (2x + 5)$\sqrt {{{\left( {2{\rm{x}} + 5} \right)}^2} - 4}
$ – 4log.(2x + 5 + $\sqrt {{{\left( {2{\rm{x}} + 5} \right)}^2} - 4} $) + c2
= (2x + 5)$\sqrt {4{{\rm{x}}^2} + 20{\rm{x}} + 21} $ –
4.log(2x + 5 + $\sqrt {4{{\rm{x}}^2} + 20{\rm{x}} + 2} $) + c2
I = I1- I2.
= $\frac{1}{6}$ (4x2 + 20x + 21)3/2 –
(2x + 5)$\sqrt {4{{\rm{x}}^2} + 20{\rm{x}} + 21} $ + 4.log $\left( {2{\rm{x}} +
5 + \sqrt {4{{\rm{x}}^2} + 20{\rm{x}} + 21} } \right)$ + c.