Find the value of x d x (1-x^2 )^ 3 / 2 : - Vidyadip

Question

Find the value of $\int \frac{x d x}{\left(1-x^2\right)^{3 / 2}}$ :

✓

Answer

$\int \frac{x d x}{\left(1-x^2\right)^{3 / 2}}$
Let
$1-x^2=t $
$\Rightarrow-2 x\ d x=d t$
$\therefore x \ d x=\frac{-1}{2} d t$
$\Rightarrow I =\frac{-1}{2} \int \frac{d t}{t^{3 / 2}}$
$=\frac{-1}{2} \int t^{-3 / 2} d t$
$=\frac{-1}{2} \frac{t^{-1 / 2}}{-1 / 2}+ C $
$=\frac{1}{\sqrt{1-x^2}}+ C \quad$

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