Question

Evaluate:

$\int \frac{1-x^3}{\left(1-2x\right)}dx$

Answer 1

$\int \frac{1-x^3}{\left(1-2x\right)}dx$

$=\int \frac{x^3-1}{2x-1}dx$

Let $u=2x-1\Rightarrow \frac{u+1}{2}=x$

$du=2dx$

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

$=\frac{1}{16}\int \frac{\left(4-1\right)\left(4^2+4u+7\right)}{4}du$

$=\frac{1}{16}\int \left(4^2+3u-\frac{7}{u}+3\right)du$

$=\frac{1}{16}\left[\frac{u^3}{3}+\frac{3u^2}{2}-7lu\left(4\right)+3u\right]$

$=\frac{{\left(2x-1\right)}^3}{48}+\frac{3{\left(2x-1\right)}^2}{32}-7lu\left(2x-1\right)+3\left(2x-1\right)+C$