What is the integral from-pi/2 to pi/2 of cos^3(θ) ?

The integral from-pi/2 to pi/2 of cos^3(θ) is 4/3

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

integral from-pi/2 to pi/2 of cos^3(θ)

\int_{- \frac{π}{2}}^{\frac{π}{2}} cos^{3} (θ) d θ

\frac{4}{3}

Decimal

1.33333 \dots

Solution steps

\int_{- \frac{π}{2}}^{\frac{π}{2}} cos^{3} (θ) d θ

Rewrite using trig identities

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

Apply u-substitution

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

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

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

= 2 - \frac{2}{3}

Simplify

2 - \frac{2}{3} : \frac{4}{3}

= \frac{4}{3}

What is the integral from-pi/2 to pi/2 of cos^3(θ) ?
The integral from-pi/2 to pi/2 of cos^3(θ) is 4/3