What is the integral from 0 to pi of sin^2(θ)sin(θ) ?

The integral from 0 to pi of sin^2(θ)sin(θ) is 4/3

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 of sin^2(θ)sin(θ)

\int_{0}^{π} sin^{2} (θ) sin (θ) d θ

\frac{4}{3}

Decimal Notation

1.33333 \dots

Solution steps

\int_{0}^{π} sin^{2} (θ) sin (θ) d θ

Rewrite using trig identities

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

Apply u-substitution

: \int_{1}^{- 1} - 1 + u^{2} d u

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

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \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}

\int_{- 3}^{2} 2 x e^{x} d x

\int_{- \infty}^{l n (7)} e^{x} d x

\int_{0}^{3} x 1 + 4 x^{2} d x

\int_{0}^{2} 3 x^{2} - 2 x + 1 d x

What is the integral from 0 to pi of sin^2(θ)sin(θ) ?
The integral from 0 to pi of sin^2(θ)sin(θ) is 4/3