What is the integral from 0 to pi/2 of (sin(x))^4 ?

The integral from 0 to pi/2 of (sin(x))^4 is (3pi)/(16)

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integral from 0 to pi/2 of (sin(x))^4

\int_{0}^{\frac{π}{2}} (sin (x))^{4} d x

\frac{3 π}{16}

Decimal

0.58904 \dots

Solution steps

\int_{0}^{\frac{π}{2}} (sin (x))^{4} d x

Simplify

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

Apply Integral Reduction

= - [\frac{cos ( x ) sin ^{3} ( x )}{4}]_{0}^{\frac{π}{2}} + \frac{3}{4} \cdot \int_{0}^{\frac{π}{2}} sin^{2} (x) d x

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

= - [\frac{cos ( x ) sin ^{3} ( x )}{4}]_{0}^{\frac{π}{2}} + \frac{3}{4} \cdot \frac{π}{4}

Simplify

- [\frac{cos ( x ) sin ^{3} ( x )}{4}]_{0}^{\frac{π}{2}} + \frac{3}{4} \cdot \frac{π}{4} : - [\frac{1}{4} sin^{3} (x) cos (x)]_{0}^{\frac{π}{2}} + \frac{3 π}{16}

= - [\frac{1}{4} sin^{3} (x) cos (x)]_{0}^{\frac{π}{2}} + \frac{3 π}{16}

Compute the boundaries:

0

= - 0 + \frac{3 π}{16}

Simplify

= \frac{3 π}{16}

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What is the integral from 0 to pi/2 of (sin(x))^4 ?
The integral from 0 to pi/2 of (sin(x))^4 is (3pi)/(16)