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

The integral from pi/3 to pi/2 of cos^3(x) is 2/3-(3sqrt(3))/8

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

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

\frac{2}{3} - \frac{3 3}{8}

Decimal Notation

0.01714 \dots

Solution steps

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

Rewrite using trig identities

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

Apply u-substitution

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

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

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

\int_{\frac{3}{2}}^{1} u^{2} d u = \frac{1}{3} - \frac{3}{8}

= 1 - \frac{3}{2} - (\frac{1}{3} - \frac{3}{8})

Simplify

1 - \frac{3}{2} - (\frac{1}{3} - \frac{3}{8}) : \frac{2}{3} - \frac{3 3}{8}

= \frac{2}{3} - \frac{3 3}{8}

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What is the integral from pi/3 to pi/2 of cos^3(x) ?
The integral from pi/3 to pi/2 of cos^3(x) is 2/3-(3sqrt(3))/8