What is the integral from 0 to pi/2 of cos^{13}(x) ?

The integral from 0 to pi/2 of cos^{13}(x) is 1024/3003

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integral from 0 to pi/2 of cos^{13}(x)

\int_{0}^{\frac{π}{2}} cos^{13} (x) d x

\frac{1024}{3003}

Decimal Notation

0.34099 \dots

Solution steps

\int_{0}^{\frac{π}{2}} cos^{13} (x) d x

Rewrite using trig identities

= \int_{0}^{\frac{π}{2}} (1 - sin^{2} (x))^{6} cos (x) d x

Apply u-substitution

: \int_{0}^{1} (1 - u^{2})^{6} d u

= \int_{0}^{1} (1 - u^{2})^{6} d u

Expand

(1 - u^{2})^{6} : 1 - 6 u^{2} + 15 u^{4} - 20 u^{6} + 15 u^{8} - 6 u^{10} + u^{12}

= \int_{0}^{1} 1 - 6 u^{2} + 15 u^{4} - 20 u^{6} + 15 u^{8} - 6 u^{10} + u^{12} 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} 6 u^{2} d u + \int_{0}^{1} 15 u^{4} d u - \int_{0}^{1} 20 u^{6} d u + \int_{0}^{1} 15 u^{8} d u - \int_{0}^{1} 6 u^{10} d u + \int_{0}^{1} u^{12} d u

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

\int_{0}^{1} 6 u^{2} d u = 2

\int_{0}^{1} 15 u^{4} d u = 3

\int_{0}^{1} 20 u^{6} d u = \frac{20}{7}

\int_{0}^{1} 15 u^{8} d u = \frac{5}{3}

\int_{0}^{1} 6 u^{10} d u = \frac{6}{11}

\int_{0}^{1} u^{12} d u = \frac{1}{13}

= 1 - 2 + 3 - \frac{20}{7} + \frac{5}{3} - \frac{6}{11} + \frac{1}{13}

Simplify

1 - 2 + 3 - \frac{20}{7} + \frac{5}{3} - \frac{6}{11} + \frac{1}{13} : \frac{1024}{3003}

= \frac{1024}{3003}

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What is the integral from 0 to pi/2 of cos^{13}(x) ?
The integral from 0 to pi/2 of cos^{13}(x) is 1024/3003