What is the integral from 0 to pi/6 of cos^3(x) ?

The integral from 0 to pi/6 of cos^3(x) is 11/24

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integral from 0 to pi/6 of cos^3(x)

\int_{0}^{\frac{π}{6}} cos^{3} (x) d x

\frac{11}{24}

Decimal

0.45833 \dots

Solution steps

\int_{0}^{\frac{π}{6}} cos^{3} (x) d x

Rewrite using trig identities

= \int_{0}^{\frac{π}{6}} (1 - sin^{2} (x)) cos (x) d x

Apply u-substitution

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

= \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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\int_{0}^{\frac{π}{2}} 11 sin^{2} (2 x) d x

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What is the integral from 0 to pi/6 of cos^3(x) ?
The integral from 0 to pi/6 of cos^3(x) is 11/24