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

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

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

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

\frac{3 π}{16}

Decimal Notation

0.58904 \dots

Solution steps

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

Apply Integral Reduction

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

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

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

Simplify

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

= - [\frac{1}{4} sin^{3} (x) cos (x)]_{\frac{π}{2}}^{π} + \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 pi/2 to pi of sin^4(x) ?
The integral from pi/2 to pi of sin^4(x) is (3pi)/(16)