The integral (^2x^2x)/((^5x + ^3x^2x + ^3x^2x + ^5x)^2),dx is equal to:

Question

The integral $\displaystyle\int \frac{\sin^2x\cos^2x}{(\sin^5x + \cos^3x\sin^2x + \sin^3x\cos^2x + \cos^5x)^2}\,dx$ is equal to:

Accepted Answer

Let $I = \int \dfrac{\sin^2x\cos^2x}{[(\sin^2x+\cos^2x)(\sin^3x+\cos^3x)]^2}\,dx = \int \dfrac{\sin^2x\cos^2x}{(\sin^3x+\cos^3x)^2}\,dx = \int \dfrac{	an^2x\sec^2x}{(1+	an^3x)^2}\,dx$

Put $1+	an^3x = t \Rightarrow 3	an^2x\sec^2x\,dx = dt$

$I = \frac{1}{3}\int \frac{dt}{t^2} = -\frac{1}{3t}+C = \frac{-1}{3(1+	an^3x)}+C$

Solutions · 1