What is the integral from 0 to pi/6 of sin^4(6x) ?

The integral from 0 to pi/6 of sin^4(6x) is pi/(16)

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integral from 0 to pi/6 of sin^4(6x)

\int_{0}^{\frac{π}{6}} sin^{4} (6 x) d x

\frac{π}{16}

Decimal

0.19634 \dots

Solution steps

\int_{0}^{\frac{π}{6}} sin^{4} (6 x) d x

Apply u-substitution

= \int_{0}^{π} sin^{4} (u) \frac{1}{6} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{6} \cdot \int_{0}^{π} sin^{4} (u) d u

Apply Integral Reduction

= \frac{1}{6} (- [\frac{cos ( u ) sin ^{3} ( u )}{4}]_{0}^{π} + \frac{3}{4} \cdot \int_{0}^{π} sin^{2} (u) d u)

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

= \frac{1}{6} (- [\frac{cos ( u ) sin ^{3} ( u )}{4}]_{0}^{π} + \frac{3}{4} \cdot \frac{π}{2})

Simplify

\frac{1}{6} (- [\frac{cos ( u ) sin ^{3} ( u )}{4}]_{0}^{π} + \frac{3}{4} \cdot \frac{π}{2}) : \frac{1}{6} (- [\frac{1}{4} sin^{3} (u) cos (u)]_{0}^{π} + \frac{3 π}{8})

= \frac{1}{6} (- [\frac{1}{4} sin^{3} (u) cos (u)]_{0}^{π} + \frac{3 π}{8})

Compute the boundaries:

0

= \frac{1}{6} (- 0 + \frac{3 π}{8})

Simplify

= \frac{π}{16}

What is the integral from 0 to pi/6 of sin^4(6x) ?
The integral from 0 to pi/6 of sin^4(6x) is pi/(16)