What is the integral of sin^7(x/4)cos(x/4) ?

The integral of sin^7(x/4)cos(x/4) is 1/2 sin^8(x/4)+C

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integral of sin^7(x/4)cos(x/4)

\int sin^{7} (\frac{x}{4}) cos (\frac{x}{4}) d x

\frac{1}{2} sin^{8} (\frac{x}{4}) + C

Solution steps

\int sin^{7} (\frac{x}{4}) cos (\frac{x}{4}) d x

Apply u-substitution

Take the constant out:

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

Apply the Power Rule

: \frac{7}{8} u^{\frac{8}{7}}

= \frac{4}{7} \cdot \frac{7}{8} u^{\frac{8}{7}}

Substitute back

u = sin^{7} (\frac{x}{4})

= \frac{4}{7} \cdot \frac{7}{8} (sin^{7} (\frac{x}{4}))^{\frac{8}{7}}

Simplify

\frac{4}{7} \cdot \frac{7}{8} (sin^{7} (\frac{x}{4}))^{\frac{8}{7}} : \frac{1}{2} sin^{8} (\frac{x}{4})

= \frac{1}{2} sin^{8} (\frac{x}{4})

Add a constant to the solution

= \frac{1}{2} sin^{8} (\frac{x}{4}) + C

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What is the integral of sin^7(x/4)cos(x/4) ?
The integral of sin^7(x/4)cos(x/4) is 1/2 sin^8(x/4)+C