Question

Solve $\int \left(\frac{x\sqrt{1+x}}{\sqrt{1-x}}\right)dx$

Answer 1

$\int \frac{x\sqrt{1+x}}{\sqrt{1-x}}dx$

This can be written as

$\Rightarrow \int x\sqrt{\frac{(1+x)(1+x)}{(1-x)(1+x)}}dx$

$\Rightarrow \int x\sqrt{\frac{(1+x{)}^2}{1-x^2}}dx$

$\Rightarrow \int x\frac{(1+x)}{\sqrt{1-x^2}}dx$

$\Rightarrow \int \frac{(x+x^2)}{\sqrt{1-x^2}}dx$

$\Rightarrow \int \frac{x}{\sqrt{1-x^2}}dx+\int \frac{x^2}{\sqrt{1-x^2}}dx$

Take $x=sin\theta \Rightarrow dx=cos\theta d\theta$

$\Rightarrow \int \frac{sin\theta cos\theta }{\sqrt{1-si{n}^2\theta }}d\theta +\int \frac{si{n}^2\theta cos\theta }{\sqrt{1-si{n}^2\theta }}d\theta$

$\Rightarrow \int \frac{sin\theta cos\theta }{cos\theta }d\theta +\int \frac{si{n}^2\theta cos\theta }{cos\theta }d\theta$

$\Rightarrow \int sin\theta d\theta +\int si{n}^2\theta d\theta$

$\Rightarrow \int sin\theta d\theta +\frac{1}{2}\int 2si{n}^2\theta d\theta$

$\Rightarrow -cos\theta +\frac{1}{2}\int (1-cos2\theta )d\theta$

$\Rightarrow -cos\theta +\frac{1}{2}\int 1d\theta -\frac{1}{2}\int cos2\theta d\theta$

$\Rightarrow -cos\theta +\frac{1}{2}\theta -\frac{1}{4}sin2\theta$

$\Rightarrow -\sqrt{1-x^2}+\frac{1}{2}si{n}^{-1}x-\frac{1}{2}x\sqrt{1-x^2}$

$\Rightarrow -\sqrt{1-x^2}(1+\frac{x}{2})+\frac{1}{2}si{n}^{-1}x$