Question

Evaluate the integrals using substitution.
${\int }_0^1\frac{x^2}{x^2+1}dx$.

Answer 1

${\int }_0^{12}\frac{x^2}{x^2+1}dx$

$x=\tan \theta \Rightarrow dx={\sec }^2\theta d\theta$

$x=0\Rightarrow \theta =0^o,x=1\Rightarrow \theta =\frac{\pi }{4}$

$={\int }_0^{\pi /4}\frac{{\tan }^2\theta }{{\sec }^2\theta }.{\sec }^2\theta .d\theta ={\int }_0^{\pi /4}{\tan }^2\theta .d\theta$

$={\int }_0^{\pi /4}({\sec }^2\theta -1).d\theta ={[\tan \theta -\theta ]}_0^{\pi /4}=1-\frac{\pi }{4}$