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

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

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

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

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

Solution steps

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

Rewrite using trig identities

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

Take the constant out:

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

= \frac{1}{2} \cdot \int 1 - cos (3 x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int 1 d x = x

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

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

Add a constant to the solution

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

y = \frac{4}{x}

\int 12 (x - 4)^{5} d x

9 x - 36

\frac{\partial}{\partial x} (- 4 x + 5 - 6 x y)

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