What is the integral of sin^2(nx) ?

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

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integral of sin^2(nx)

\int sin^{2} (n x) d x

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

Solution steps

\int sin^{2} (n x) d x

Rewrite using trig identities

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

\int 1 d x = x

\int cos (2 n x) d x = \frac{1}{2 n} sin (2 n x)

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

Add a constant to the solution

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

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