Question

$\int \frac{dx}{{\cos }^{3}x\sqrt{\left(\sin 2x\right)}}=\frac{\sqrt{2}}{5}(t^2+5)\sqrt{t}$. where $t=\tan x$.

If this is true enter 1, else enter 0.

Answer 1

Let $I=\int \frac{dx}{{\cos }^{3}x\sqrt{\left(\sin 2x\right)}}$
Put $\sin 2x=\frac{2\tan x}{1+{\tan }^{2}x}$
$I=\int \frac{{\sec }^{2}x.{\sec }^{2}x}{\sqrt{\left(2\tan x\right)}}dx=\int \frac{1}{\sqrt{2}}\frac{(1+t^2)}{\sqrt{t}}dt$
$=\frac{1}{\sqrt{2}}\int (\frac{1}{\sqrt{t}}+t^{3/2})dt=\frac{\sqrt{2}}{5}(t^2+5)\sqrt{t}$