What is the integral from 0 to pi/2 of (cos(x))^5 ?

The integral from 0 to pi/2 of (cos(x))^5 is 8/15

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integral from 0 to pi/2 of (cos(x))^5

\int_{0}^{\frac{π}{2}} (cos (x))^{5} d x

\frac{8}{15}

Decimal

0.53333 \dots

Solution steps

\int_{0}^{\frac{π}{2}} (cos (x))^{5} d x

Simplify

= \int_{0}^{\frac{π}{2}} cos^{5} (x) d x

Rewrite using trig identities

= \int_{0}^{\frac{π}{2}} (1 - sin^{2} (x))^{2} cos (x) d x

Apply u-substitution

: \int_{0}^{1} (1 - u^{2})^{2} d u

= \int_{0}^{1} (1 - u^{2})^{2} d u

Expand

(1 - u^{2})^{2} : 1 - 2 u^{2} + u^{4}

= \int_{0}^{1} 1 - 2 u^{2} + u^{4} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{0}^{1} 1 d u - \int_{0}^{1} 2 u^{2} d u + \int_{0}^{1} u^{4} d u

\int_{0}^{1} 1 d u = 1

\int_{0}^{1} 2 u^{2} d u = \frac{2}{3}

\int_{0}^{1} u^{4} d u = \frac{1}{5}

= 1 - \frac{2}{3} + \frac{1}{5}

Simplify

1 - \frac{2}{3} + \frac{1}{5} : \frac{8}{15}

= \frac{8}{15}

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What is the integral from 0 to pi/2 of (cos(x))^5 ?
The integral from 0 to pi/2 of (cos(x))^5 is 8/15