What is the integral from 0 to pi of sin^3(φ) ?

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

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integral from 0 to pi of sin^3(φ)

\int_{0}^{π} sin^{3} (φ) d φ

\frac{4}{3}

Decimal

1.33333 \dots

Solution steps

\int_{0}^{π} sin^{3} (φ) d φ

Rewrite using trig identities

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

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_{0}^{\frac{2}{3}} 1 + 36 x d x

\int_{- 3}^{3} x^{3} + \frac{1}{x} d x

\int_{0}^{\frac{1}{4}} y arctan (4 y) d y

\int_{0}^{\frac{π}{2}} sin^{2} (2 n x) d x

\int_{- \frac{1}{2}}^{\frac{1}{2}} x arctan (x) d x

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