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

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

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integral of cos^2(nx)

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

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

Solution steps

\int cos^{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

\int_{0}^{1} (x^{2} + 7) e^{- x} d x

16 y^{^{''}} + 16 y^{^{'}} - 5 y = 0

f (x) = 2 x

\int \frac{x + 8}{x ^{2} + 25} d x

\frac{d}{d x} (x^{2} e^{x^{2}})

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