Evaluate the following integral: ∫(xsin x cos x)/(sin^4 x + cos^4 x)dx, x ∈ [0, π/2]

Question

Evaluate the following integral:

\(\int\limits_{0}^{\pi/2}\cfrac{\text xsin\text cos\text x}{sin^4\text x+cos^4\text x}d\text x \)

Answer 1

Let us assume

I = \(\int\limits_{0}^{\pi/2}\cfrac{\text xsin\text cos\text x}{sin^4\text x+cos^4\text x}d\text x \)....equation 1

By property, we know that

Adding equation 1 and 2 

Thus

Let tan²x = y

Differentiating both sides

2 tanx sec²xdx = dy

substituting values in equation 3