Check the solution to "Trigonometric integrals Evaluate the following integrals. ∫sin5xdx"

Tammy Todd

Answered question

2021-11-10

Trigonometric integrals Evaluate the following integrals.

\int \sin^{5} x d x

Laith Petty

Skilled2021-11-11Added 103 answers

Step 1
Evaluate

\int \sin^{5} x d x

Step 2
let

I = \int \sin^{5} x d x

I = \int \sin^{4} x \sin x d x

I = \int {(\sin^{2} x)}^{2} \sin x d x

I = \int {(1 - \cos^{2} x)}^{2} \sin x d x

put

\cos x = u

on differentiation we have

- \sin x d x = d u

I = \int {(1 - u^{2})}^{2} (- d u)

I = - \int (1 - 2 u^{2} + u^{4}) d u

I = - [u - \frac{2 u^{3}}{3} + \frac{u^{5}}{5}]

I = [\frac{2 u^{3}}{3} - u - \frac{u^{5}}{5}]

I = \frac{2 \cos^{3} x}{3} - \cos x - \frac{\cos^{5} x}{5}

I = \frac{2 \cos^{3} x}{3} - \cos x - \frac{\cos^{5} x}{5} + c

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