What is the integral of sin^3(x/5)cos(x/5) ?

The integral of sin^3(x/5)cos(x/5) is 5/4 sin^4(x/5)+C

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integral of sin^3(x/5)cos(x/5)

\int sin^{3} (\frac{x}{5}) cos (\frac{x}{5}) d x

\frac{5}{4} sin^{4} (\frac{x}{5}) + C

Solution steps

\int sin^{3} (\frac{x}{5}) cos (\frac{x}{5}) 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{3}{4} u^{\frac{4}{3}}

= \frac{5}{3} \cdot \frac{3}{4} u^{\frac{4}{3}}

Substitute back

u = sin^{3} (\frac{x}{5})

= \frac{5}{3} \cdot \frac{3}{4} (sin^{3} (\frac{x}{5}))^{\frac{4}{3}}

Simplify

\frac{5}{3} \cdot \frac{3}{4} (sin^{3} (\frac{x}{5}))^{\frac{4}{3}} : \frac{5}{4} sin^{4} (\frac{x}{5})

= \frac{5}{4} sin^{4} (\frac{x}{5})

Add a constant to the solution

= \frac{5}{4} sin^{4} (\frac{x}{5}) + C

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What is the integral of sin^3(x/5)cos(x/5) ?
The integral of sin^3(x/5)cos(x/5) is 5/4 sin^4(x/5)+C