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

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

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

\int_{0}^{\frac{π}{2}} 15 cos^{5} (x) d x

8

Solution steps

\int_{0}^{\frac{π}{2}} 15 cos^{5} (x) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 15 \cdot \int_{0}^{\frac{π}{2}} cos^{5} (x) d x

Rewrite using trig identities

= 15 \cdot \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

= 15 \cdot \int_{0}^{1} (1 - u^{2})^{2} d u

Expand

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

= 15 \cdot \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

= 15 (\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}

= 15 (1 - \frac{2}{3} + \frac{1}{5})

Simplify

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

= 8

x \to - 2 + lim (- \frac{1}{x ^{2} - 4})

\frac{d}{d x} (- a x + \frac{d}{x ^{2}})

\frac{s}{( s ^{2} + 1 ) ( s ^{2} - 2 s + 2 )}

\int \frac{x - 5}{x ^{2} + 9} d x

\int x^{2} sin (2 A x) d x

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