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

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

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

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

- \frac{2}{3}

Decimal Notation

- 0.66666 \dots

Solution steps

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

Rewrite using trig identities

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

Apply u-substitution

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

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

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

= - \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})

Join

1 - \frac{1}{3} : \frac{2}{3}

= - \frac{2}{3}

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