Integral of $\int \frac{x \sin{x} \cos{x}}{(a^2 \sin^2 x + b^2 \cos^2 x)^2} dx $

Question

Please help me to find the integral of

$$\int \frac{x \sin{x} \cos{x}}{(a^2 \sin^2 x + b^2 \cos^2 x)^2} dx $$

There is a problem in having x in the numerator. Please guide me . Thanks in advance.

Answer 1

Observe that $$\dfrac{d(a^2\sin^2x+b^2\cos^2x)}{dx}=2(a^2-b^2)\sin x\cos x$$

Integrate by parts as $$\int x\dfrac{\sin x\cos x}{(a^2\sin^2x+b^2\cos^2x)^2}dx$$

$$=x\int\dfrac{\sin x\cos x}{(a^2\sin^2x+b^2\cos^2x)^2}dx-\int\left[\frac{dx}{dx}\cdot \int\dfrac{\sin x\cos x}{(a^2\sin^2x+b^2\cos^2x)^2}dx\right]dx$$

Finally, $\int\dfrac{dx}{a^2\sin^2x+b^2\cos^2x}=\dfrac1{a^2}\int\dfrac{\sec^2x\ dx}{\tan^2x+(b/a)^2}$