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

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

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integral from 0 to pi/2 of sin^3(φ)

\int_{0}^{\frac{π}{2}} sin^{3} (φ) d φ

\frac{2}{3}

Decimal

0.66666 \dots

Solution steps

\int_{0}^{\frac{π}{2}} sin^{3} (φ) d φ

Rewrite using trig identities

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

Apply u-substitution

: \int_{1}^{0} - 1 + u^{2} d u

= \int_{1}^{0} - 1 + u^{2} d u

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

= - \int_{0}^{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_{0}^{1} 1 d u + \int_{0}^{1} u^{2} d u)

\int_{0}^{1} 1 d u = 1

\int_{0}^{1} u^{2} d u = \frac{1}{3}

= - (- 1 + \frac{1}{3})

Simplify

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

= \frac{2}{3}

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