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

The integral from-pi/2 to pi/2 of cos^5(x) is 16/15

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

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

\frac{16}{15}

Decimal

1.06666 \dots

Solution steps

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

Rewrite using trig identities

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

Apply u-substitution

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

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

Expand

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

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

\int_{- 1}^{1} 1 d u = 2

\int_{- 1}^{1} 2 u^{2} d u = \frac{4}{3}

\int_{- 1}^{1} u^{4} d u = \frac{2}{5}

= 2 - \frac{4}{3} + \frac{2}{5}

Simplify

2 - \frac{4}{3} + \frac{2}{5} : \frac{16}{15}

= \frac{16}{15}

\int_{- 1}^{0} 2 x d x

\int_{0}^{\frac{π}{2}} cos (n x) d x

\int_{0}^{\frac{24}{25}} 1 + 25 x d x

\int_{1}^{4} e^{- x} (x - 1) d x

\int_{2}^{3} \frac{2 x}{( x ^{2} + 3 ) ^{3}} d x

What is the integral from-pi/2 to pi/2 of cos^5(x) ?
The integral from-pi/2 to pi/2 of cos^5(x) is 16/15