Question

Solve:
$\int \frac{sin2x}{\sqrt{9-si{n}^{4}x}}dx$

Answer 1

We have,

$\int \frac{\sin 2x}{\sqrt{9-{\sin }^{4}x}}dx$

Now,

$\int \frac{\sin 2x}{\sqrt{3^2-{\left({\sin }^{2}x\right)}^2}}dx$

Let

${\sin }^{2}x=t$

$\frac{d}{dx}{\sin }^{2}x=dt$

$2\sin x\cos xdx=dt$

$\sin 2xdx=dt$

Now,

$\int \frac{dt}{\sqrt{3^2-t^2}}$

Using that,

$\int \frac{1}{\sqrt{a^2-x^2}}dx={\sin }^{-1}\frac{x}{a}$

$\int \frac{dt}{\sqrt{3^2-t^2}}={\sin }^{-1}\frac{t}{3}+C$

$\int \frac{dt}{\sqrt{3^2-t^2}}={\sin }^{-1}\frac{{\sin }^{2}x}{3}+C$

Hence, this is the answer.