What is the integral from 0 to pi/2 of cos^7(x) ?

The integral from 0 to pi/2 of cos^7(x) is 16/35

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integral from 0 to pi/2 of cos^7(x)

\int_{0}^{\frac{π}{2}} cos^{7} (x) d x

\frac{16}{35}

Decimal Notation

0.45714 \dots

Solution steps

\int_{0}^{\frac{π}{2}} cos^{7} (x) d x

Rewrite using trig identities

= \int_{0}^{\frac{π}{2}} (1 - sin^{2} (x))^{3} cos (x) d x

Apply u-substitution

: \int_{0}^{1} (1 - u^{2})^{3} d u

= \int_{0}^{1} (1 - u^{2})^{3} d u

Expand

(1 - u^{2})^{3} : 1 - 3 u^{2} + 3 u^{4} - u^{6}

= \int_{0}^{1} 1 - 3 u^{2} + 3 u^{4} - u^{6} 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} 3 u^{2} d u + \int_{0}^{1} 3 u^{4} d u - \int_{0}^{1} u^{6} d u

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

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

\int_{0}^{1} 3 u^{4} d u = \frac{3}{5}

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

= 1 - 1 + \frac{3}{5} - \frac{1}{7}

Simplify

1 - 1 + \frac{3}{5} - \frac{1}{7} : \frac{16}{35}

= \frac{16}{35}

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What is the integral from 0 to pi/2 of cos^7(x) ?
The integral from 0 to pi/2 of cos^7(x) is 16/35