Evaluate the following Integral: ∫(sin x cos x)/(1 + sin^4x)dx, x ∈ [0, π/2]

Question

Evaluate the following Integral:

\(\int\limits_0^{\pi/2}\cfrac{sin\text x\,cos\,\text x}{1+sin^4\text x}d\text x\)

Answer 1

Given Definite Integral can be written as:

⇒ I(x) = \(\int\limits_0^{\pi/2}\cfrac{sin\text x\,cos\,\text x}{1+sin^4\text x}d\text x\)......(1)

Let us assume, y = sin²x

Differentiating w.r.t x on both sides we get,

⇒ d(y) = d(sin²x)

⇒ dy = 2sinxcosxdx

⇒ sin x cos x dx = \(\cfrac{dy}2\)......(2)

Upper limit for the Definite Integral:

⇒ x = \(\cfrac{\pi}2\) ⇒ y = sin²\(\cfrac{\pi}2\)

⇒ y = 1......(3)

Lower limit for the Definite Integral:

⇒ x = 0 ⇒ y = sin²0

⇒ y = 0……(4)

Substituting (2),(3),(4) in the eq(1) we get,

[here f’(x) is derivative of f(x))