What is the integral of sin^6(x/2)cos(x/2) ?

The integral of sin^6(x/2)cos(x/2) is 2/7 sin^7(x/2)+C

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integral of sin^6(x/2)cos(x/2)

\int sin^{6} (\frac{x}{2}) cos (\frac{x}{2}) d x

\frac{2}{7} sin^{7} (\frac{x}{2}) + C

Solution steps

\int sin^{6} (\frac{x}{2}) cos (\frac{x}{2}) d x

Apply u-substitution

= \int sin^{6} (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^{6} (u) cos (u) d u

Apply u-substitution

= 2 \cdot \int v^{6} d v

Apply the Power Rule

= 2 \cdot \frac{v ^{7}}{7}

Substitute back

= 2 \cdot \frac{sin ^{7} ( \frac{x}{2} )}{7}

Simplify

2 \cdot \frac{sin ^{7} ( \frac{x}{2} )}{7} : \frac{2}{7} sin^{7} (\frac{x}{2})

= \frac{2}{7} sin^{7} (\frac{x}{2})

Add a constant to the solution

= \frac{2}{7} sin^{7} (\frac{x}{2}) + C

What is the integral of sin^6(x/2)cos(x/2) ?
The integral of sin^6(x/2)cos(x/2) is 2/7 sin^7(x/2)+C