Question

Show that ${\int }_0^1\frac{1}{{(1+x^2)}^{3/2}}dx=\frac{3}{\sqrt{\left(2\right)}}$

Answer 1

Let $I={\int }_0^1\frac{1}{{(1+x^2)}^{3/2}}dx$

Put $x=\tan t$

$\Rightarrow dx={\sec }^{2}tdt$

When $x=0\Rightarrow t=0$

When $x=1\Rightarrow t=\frac{\pi }{4}$

$I={\int }_0^{\frac{\pi }{4}}\frac{{\sec }^{2}t}{{\left({\sec }^{2}t\right)}^{3/2}}dt$

$\therefore I={\int }_0^{\frac{\pi }{4}}\cos tdt$

$={\left[\sin t\right]}_0^{\frac{\pi }{4}}=\frac{1}{\sqrt{2}}$