Evaluate the following Integral: ∫(sin^2x +cos^2x )/(sin^3x +cos^3x)^2 dx, x ∈ [0, π/4]

Question

Evaluate the following Integral:

\(\int\limits_{0}^{\pi/4}\cfrac{sin^2\text x\,cos^2\text x}{(sin^3\text x+cos^3\text x)^2}d\text x \)

Answer 1

Let I = \(\int\limits_{0}^{\pi/4}\cfrac{sin^2\text x\,cos^2\text x}{(sin^3\text x+cos^3\text x)^2}d\text x \)

Put tan x = t

⇒ sec²x dx = dt (Differentiating both sides)

When x = 0, t = tan 0 = 0

When x = \(\cfrac{\pi}4\), t = tan \(\cfrac{\pi}4\) = 1

So, the new limits are 0 and1.

Substituting this in the original integral,

Put t³ = u

⇒ 3t²dt = du (Differentiating both sides)

⇒ t²dt = \(\cfrac13\)du

When t = 0, u = 0³ = 0

When t = 1, u = 1³ = 1

So, the new limits are 0 and1.

Substituting this in the original integral,