Question

Integral of $(2x-3{)}^5+\frac{1}{(7x-5{)}^3}+\frac{1}{\sqrt{(5x-4)}}+\frac{1}{2-3x}+\sqrt{3x+2}$ is

Answer 1

Say, $I=\int \left(\right(2x-3{)}^5+\frac{1}{(7x-5{)}^3}+\frac{1}{\sqrt{(5x-4)}}+\frac{1}{2-3x}+\sqrt{3x+2})dx$

$=\int (2x-3{)}^{5}dx+\int \frac{1}{(7x-5{)}^3}dx+\int \frac{1}{\sqrt{(5x-4)}}dx+\int \frac{1}{2-3x}dx+\int \sqrt{3x+2}dx$

$=\frac{1}{2}\left(\frac{(2x-3{)}^{5+1}}{5+1}\right)+\frac{1}{7}\left(\frac{(7x-5{)}^{-3+1}}{-3+1}\right)+\frac{1}{5}\left(\frac{(5x-4{)}^{\frac{-1}{2}+1}}{-\frac{1}{2}+1}\right)+(-\frac{1}{3}()\log (2-3x)+\frac{1}{3}\left(\frac{(3x+2{)}^{\frac{1}{2}+1}}{\frac{1}{2}+1}\right)+C$

$\Rightarrow I=\frac{1}{2}(2x-3{)}^6-\frac{1}{14}\cdot \frac{1}{(7x-5{)}^2}+\frac{2}{5}\sqrt{5x-4}-\frac{1}{3}\log (2-3x)+\frac{2}{9}(3x+2{)}^{3/2}+C$