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

The integral from pi/3 to pi/2 of sin^3(x) is 11/24

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integral from pi/3 to pi/2 of sin^3(x)

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

\frac{11}{24}

Decimal Notation

0.45833 \dots

Solution steps

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

Rewrite using trig identities

= \int_{\frac{π}{3}}^{\frac{π}{2}} (1 - cos^{2} (x)) sin (x) d x

Apply u-substitution

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

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

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

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

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

\int_{0}^{\frac{1}{2}} u^{2} d u = \frac{1}{24}

= - (- \frac{1}{2} + \frac{1}{24})

Simplify

- (- \frac{1}{2} + \frac{1}{24}) : \frac{11}{24}

= \frac{11}{24}

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What is the integral from pi/3 to pi/2 of sin^3(x) ?
The integral from pi/3 to pi/2 of sin^3(x) is 11/24