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

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

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integral from 0 to 2pi of sin^4(θ)

\int_{0}^{2 π} sin^{4} (θ) d θ

\frac{3 π}{4}

Decimal

2.35619 \dots

Solution steps

\int_{0}^{2 π} sin^{4} (θ) d θ

Apply Integral Reduction

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

\int_{0}^{2 π} sin^{2} (θ) d θ = π

= - [\frac{cos ( θ ) sin ^{3} ( θ )}{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^4(θ) ?
The integral from 0 to 2pi of sin^4(θ) is (3pi)/4