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

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

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

\int_{0}^{2 π} (sin (x))^{4} d x

\frac{3 π}{4}

Decimal

2.35619 \dots

Solution steps

\int_{0}^{2 π} (sin (x))^{4} d x

Simplify

= \int_{0}^{2 π} sin^{4} (x) d x

Apply Integral Reduction

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

\int_{0}^{2 π} sin^{2} (x) d x = π

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

Compute the boundaries:

0

= - 0 + \frac{3}{4} π

Simplify

= \frac{3 π}{4}

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