integral of $\int \frac {dx}{2\sin(x)^2 + 3\cos(x)^2} $

Question

The answer is supposedly

$$\frac {1}{\sqrt{6}} \arctan\left(\sqrt{\frac {2}{3}}\tan x\right) + C$$

So I need to get it into form

$$\int \frac{\mathrm dx}{a^2+x^2} $$

but I am not sure what identities I need to use.

Answer 1

Hint: You can try dividing the numerator and denominator by $\cos^2 x$ and then enforcing the substitution $t=\tan x$ afterwards.

$$\int\frac {\mathrm dx}{2\sin^2 x+3\cos^2 x}=\int\mathrm dx\,\frac {\sec^2 x}{3+2\tan^2 x}=\int\frac {\mathrm dt}{3+2t^2}$$

I will let you complete the rest!