Question

Integrate the following with respect to x.

${\displaystyle \frac{8x+13}{\sqrt{4x+7}}}$

Answer 1

${\displaystyle \int \frac{8x+13}{\sqrt{4x+7}}\text{d}x=\int \frac{8x+13+1-1}{\sqrt{4x+7}} \text{d}x}$

= ${\displaystyle \int \frac{2(4x+7)-1}{\sqrt{4x+7}} \text{d}x}$

= ${\displaystyle \int (\frac{2(4x+7)}{\sqrt{4x+7}}-\frac{1}{\sqrt{4x+7}}) \text{d}x}$

= ${\displaystyle \int [2.{(4x+7)}^{\frac{1}{2}}-{(4x+7)}^{\frac{-1}{2}}]\text{d}x}$

= ${\displaystyle [\frac{2{(x+7)}^{\frac{1}{2}+1}}{(\frac{1}{2}+1)\left(4\right)}-\frac{{(4x+7)}^{\frac{-1}{2}+1}}{(\frac{-1}{2}+1)\left(4\right)}]+\text{c}}$

= ${\displaystyle \frac{2{(4x+7)}^{\frac{3}{2}}}{\left(\frac{3}{2}\right)\left(4\right)}-\frac{{(4x+7)}^{\frac{1}{2}}}{\left(\frac{1}{2}\right)\left(4\right)}+\text{c}}$

= ${\displaystyle \frac{1}{3}{(4x+7)}^{\frac{3}{2}}-\frac{1}{2}{(4x+7)}^{\frac{1}{2}}+\text{c}}$