What is the integral of sin^5(x/2)cos(x/2) ?

The integral of sin^5(x/2)cos(x/2) is 1/3 sin^6(x/2)+C

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integral of sin^5(x/2)cos(x/2)

\int sin^{5} (\frac{x}{2}) cos (\frac{x}{2}) d x

\frac{1}{3} sin^{6} (\frac{x}{2}) + C

Solution steps

\int sin^{5} (\frac{x}{2}) cos (\frac{x}{2}) d x

Apply u-substitution

: \int sin^{5} (u) cos (u) \cdot 2 d u

= \int sin^{5} (u) cos (u) \cdot 2 d u

Take the constant out:

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

= 2 \cdot \int sin^{5} (u) cos (u) d u

Apply u-substitution

: \int v^{5} d v

= 2 \cdot \int v^{5} d v

Apply the Power Rule

: \frac{v ^{6}}{6}

= 2 \cdot \frac{v ^{6}}{6}

Substitute back

= 2 \cdot \frac{sin ^{6} ( \frac{x}{2} )}{6}

Simplify

2 \cdot \frac{sin ^{6} ( \frac{x}{2} )}{6} : \frac{1}{3} sin^{6} (\frac{x}{2})

= \frac{1}{3} sin^{6} (\frac{x}{2})

Add a constant to the solution

= \frac{1}{3} sin^{6} (\frac{x}{2}) + C

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What is the integral of sin^5(x/2)cos(x/2) ?
The integral of sin^5(x/2)cos(x/2) is 1/3 sin^6(x/2)+C