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

The integral of sin^3(x/2)cos^3(x/2) is -1/3 sin^6(x/2)+1/2 sin^4(x/2)+C

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

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

- \frac{1}{3} sin^{6} (\frac{x}{2}) + \frac{1}{2} sin^{4} (\frac{x}{2}) + C

Solution steps

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

Apply Integration By Parts

: 2 sin^{3} (\frac{x}{2}) (sin (\frac{x}{2}) - \frac{1}{3} sin^{3} (\frac{x}{2})) - \int 3 sin^{3} (\frac{x}{2}) cos (\frac{x}{2}) - sin^{5} (\frac{x}{2}) cos (\frac{x}{2}) d x

= 2 sin^{3} (\frac{x}{2}) (sin (\frac{x}{2}) - \frac{1}{3} sin^{3} (\frac{x}{2})) - \int 3 sin^{3} (\frac{x}{2}) cos (\frac{x}{2}) - sin^{5} (\frac{x}{2}) cos (\frac{x}{2}) d x

\int 3 sin^{3} (\frac{x}{2}) cos (\frac{x}{2}) - sin^{5} (\frac{x}{2}) cos (\frac{x}{2}) d x = \frac{3}{2} sin^{4} (\frac{x}{2}) - \frac{1}{3} sin^{6} (\frac{x}{2})

= 2 sin^{3} (\frac{x}{2}) (sin (\frac{x}{2}) - \frac{1}{3} sin^{3} (\frac{x}{2})) - (\frac{3}{2} sin^{4} (\frac{x}{2}) - \frac{1}{3} sin^{6} (\frac{x}{2}))

Simplify

2 sin^{3} (\frac{x}{2}) (sin (\frac{x}{2}) - \frac{1}{3} sin^{3} (\frac{x}{2})) - (\frac{3}{2} sin^{4} (\frac{x}{2}) - \frac{1}{3} sin^{6} (\frac{x}{2})) : - \frac{1}{3} sin^{6} (\frac{x}{2}) + \frac{1}{2} sin^{4} (\frac{x}{2})

= - \frac{1}{3} sin^{6} (\frac{x}{2}) + \frac{1}{2} sin^{4} (\frac{x}{2})

Add a constant to the solution

= - \frac{1}{3} sin^{6} (\frac{x}{2}) + \frac{1}{2} sin^{4} (\frac{x}{2}) + C

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What is the integral of sin^3(x/2)cos^3(x/2) ?
The integral of sin^3(x/2)cos^3(x/2) is -1/3 sin^6(x/2)+1/2 sin^4(x/2)+C