Evaluate the following Integral: ∫√(sin ɸ) cos^5 ɸ dɸ, ɸ ∈ [0, π/2]

Question

Evaluate the following Integral:

\(\int\limits_0^{π/2}\sqrt{sin\,\phi} \) cos⁵ ɸ dɸ

Answer 1

Given Definite Integral can be written as:

⇒ I(x) =  \(\int\limits_0^{π/2}\sqrt{sin\,\phi} \) cos⁵ ɸ dɸ

⇒ I(x) = \(\int\limits_0^{π/2}\sqrt{sin\,\phi} \) cos⁴ɸ dɸ

Let us assume sinϕ = t,

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

⇒ d(sinϕ) = d(t)

⇒ dt = cosϕ dϕ……(2)

Upper limit for the Definite Integral:

⇒ ϕ = \(\cfrac{\pi}2\) ⇒ t = sin \((\cfrac{\pi}2)\)

⇒ t = 1....(3)

Lower limit for the Definite Integral:

⇒ ϕ=0 ⇒ t = sin(0)

⇒ t = 0……(4)

We know that cos²ϕ = 1-sin²ϕ

⇒ cos²ϕ = 1 – t²……(5)

Substituting (2),(3),(4),(5) in the eq(1), we get,

We know that:

We know that:

[here f’(x) is derivative of f(x))