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

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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

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

\frac{8}{15}

Decimal

0.53333 \dots

Solution steps

\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}

2 x^{4} + \frac{3 x}{5}, at x = 0

\int (2 + 2 x) e^{(4 x + 2 x^{2})} d x

\int_{0}^{1} \frac{4 x ^{3}}{1 + x ^{4}} d x

8 y^{^{''}} + 34 y = 0, y (0) = 2, y^{^{'}} (0) = 7

\int - 4 y d y

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