What is the integral from 0 to pi/4 of cos^3(x) ?

The integral from 0 to pi/4 of cos^3(x) is 5/(6sqrt(2))

integral from 0 to pi/4 of cos^3(x)

\int_{0}^{\frac{π}{4}} cos^{3} (x) d x

\frac{5}{6 2}

Decimal

0.58925 \dots

Solution steps

\int_{0}^{\frac{π}{4}} cos^{3} (x) d x

Rewrite using trig identities

= \int_{0}^{\frac{π}{4}} (1 - sin^{2} (x)) cos (x) d x

Apply u-substitution

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

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

\int_{0}^{\frac{2}{2}} u^{2} d u = \frac{1}{6 2}

= \frac{1}{2} - \frac{1}{6 2}

Simplify

\frac{1}{2} - \frac{1}{6 2} : \frac{5}{6 2}

= \frac{5}{6 2}

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What is the integral from 0 to pi/4 of cos^3(x) ?
The integral from 0 to pi/4 of cos^3(x) is 5/(6sqrt(2))