What is the integral of sin(3/2)xsin(1/2)x ?

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

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integral of sin(3/2)xsin(1/2)x

\int sin (\frac{3}{2}) x sin (\frac{1}{2}) x d x

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

Solution steps

\int sin (\frac{3}{2}) x sin (\frac{1}{2}) x d x

Take the constant out:

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

= sin (\frac{3}{2}) sin (\frac{1}{2}) \cdot \int xx d x

Simplify

xx : x^{2}

= sin (\frac{3}{2}) sin (\frac{1}{2}) \cdot \int x^{2} d x

Apply the Power Rule

: \frac{x ^{3}}{3}

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

Simplify

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

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

Add a constant to the solution

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

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