What is the integral of cos^6(1/2 x)sin(1/2 x) ?

The integral of cos^6(1/2 x)sin(1/2 x) is -2/7 cos^7(x/2)+C

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integral of cos^6(1/2 x)sin(1/2 x)

\int cos^{6} (\frac{1}{2} x) sin (\frac{1}{2} x) d x

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

Solution steps

\int cos^{6} (\frac{1}{2} x) sin (\frac{1}{2} x) d x

Apply u-substitution

: \int cos^{6} (u) sin (u) \cdot 2 d u

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

Apply u-substitution

: \int - v^{6} d v

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

Take the constant out:

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

= 2 (- \int v^{6} d v)

Apply the Power Rule

: \frac{v ^{7}}{7}

= 2 (- \frac{v ^{7}}{7})

Substitute back

= 2 (- \frac{cos ^{7} ( \frac{1}{2} x )}{7})

Simplify

2 (- \frac{cos ^{7} ( \frac{1}{2} x )}{7}) : - \frac{2}{7} cos^{7} (\frac{x}{2})

= - \frac{2}{7} cos^{7} (\frac{x}{2})

Add a constant to the solution

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

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What is the integral of cos^6(1/2 x)sin(1/2 x) ?
The integral of cos^6(1/2 x)sin(1/2 x) is -2/7 cos^7(x/2)+C