Check the solution to "Evaluate ∫sin5xdx."

e1s2kat26

Answered question

2021-10-07

Evaluate

\int \sin^{5} x d x

Answer & Explanation

SchulzD

Skilled2021-10-08Added 83 answers

Step 1
In some cases integrals can be simplified by use of a substitution. This usually results in simplifying the integral in terms of known standard integrals.
For the given integral the integrand can be written as

\sin (x) \sin^{4} (x) = \sin (x) {(1 - \cos^{2} (x))}^{2}

.
Then the substitution

u = \cos (x)

will simplify the integrand expression.
Step 2
The given indefinite integral to be evaluated is

\int \sin^{5} (x) d x

. This can be rewritten as

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

.Use the substitution

u = \cos (x)

. Differentiating this gives

- d u = \sin (x) d x

.Substitute this in the integral and evaluate the integral.

\int \sin^{5} (x) d x = \int \sin (x) {(1 - \cos^{2} (x))}^{2} d x

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

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

= - u - \frac{u^{5}}{5} + \frac{2 u^{3}}{3} + C

= - \cos (x) - \frac{\cos^{5} (x)}{5} + \frac{2 \cos^{3} (x)}{3} + C

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