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

The integral from 0 to pi/3 of cos^3(x) is (3sqrt(3))/8

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

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

\frac{3 3}{8}

Decimal

0.64951 \dots

Solution steps

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

Rewrite using trig identities

= \int_{0}^{\frac{π}{3}} (1 - sin^{2} (x)) cos (x) d x

Apply u-substitution

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

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

\int_{0}^{\frac{3}{2}} u^{2} d u = \frac{3}{8}

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

Simplify

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

= \frac{3 3}{8}

What is the integral from 0 to pi/3 of cos^3(x) ?
The integral from 0 to pi/3 of cos^3(x) is (3sqrt(3))/8