What is the integral of sin^3(x/2)cos(x/2) ?

The integral of sin^3(x/2)cos(x/2) is 1/2 sin^4(x/2)+C

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integral of sin^3(x/2)cos(x/2)

\int sin^{3} (\frac{x}{2}) cos (\frac{x}{2}) d x

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

Solution steps

\int sin^{3} (\frac{x}{2}) cos (\frac{x}{2}) 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{2}{3} \cdot \frac{3}{4} u^{\frac{4}{3}}

Substitute back

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

= \frac{2}{3} \cdot \frac{3}{4} (sin^{3} (\frac{x}{2}))^{\frac{4}{3}}

Simplify

\frac{2}{3} \cdot \frac{3}{4} (sin^{3} (\frac{x}{2}))^{\frac{4}{3}} : \frac{1}{2} sin^{4} (\frac{x}{2})

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

Add a constant to the solution

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

\frac{e ^{- 3 s}}{s ^{2} + 6 s + 13}

f (x) = 13

\int \frac{1 + u}{- 2 u - 1 - u ^{2}} d u

e^{- t} cos^{2} (t)

\int \frac{x ^{2} + 8 x - 3}{x ^{3} + 3 x} d x

What is the integral of sin^3(x/2)cos(x/2) ?
The integral of sin^3(x/2)cos(x/2) is 1/2 sin^4(x/2)+C