What is the integral of sin^2(x/4)cos(x/4) ?

The integral of sin^2(x/4)cos(x/4) is 4/3 sin^3(x/4)+C

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integral of sin^2(x/4)cos(x/4)

\int sin^{2} (\frac{x}{4}) cos (\frac{x}{4}) d x

\frac{4}{3} sin^{3} (\frac{x}{4}) + C

Solution steps

\int sin^{2} (\frac{x}{4}) cos (\frac{x}{4}) d x

Apply u-substitution

: \int sin^{2} (u) cos (u) \cdot 4 d u

= \int sin^{2} (u) cos (u) \cdot 4 d u

Take the constant out:

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

= 4 \cdot \int sin^{2} (u) cos (u) d u

Apply u-substitution

: \int v^{2} d v

= 4 \cdot \int v^{2} d v

Apply the Power Rule

: \frac{v ^{3}}{3}

= 4 \cdot \frac{v ^{3}}{3}

Substitute back

= 4 \cdot \frac{sin ^{3} ( \frac{x}{4} )}{3}

Simplify

4 \cdot \frac{sin ^{3} ( \frac{x}{4} )}{3} : \frac{4}{3} sin^{3} (\frac{x}{4})

= \frac{4}{3} sin^{3} (\frac{x}{4})

Add a constant to the solution

= \frac{4}{3} sin^{3} (\frac{x}{4}) + C

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What is the integral of sin^2(x/4)cos(x/4) ?
The integral of sin^2(x/4)cos(x/4) is 4/3 sin^3(x/4)+C