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

The integral from 0 to pi/2 of (cos(x))^3 is 2/3

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integral from 0 to pi/2 of (cos(x))^3

\int_{0}^{\frac{π}{2}} (cos (x))^{3} d x

\frac{2}{3}

Decimal

0.66666 \dots

Solution steps

\int_{0}^{\frac{π}{2}} (cos (x))^{3} d x

Simplify

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

Rewrite using trig identities

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

Apply u-substitution

= \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}

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