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

The integral of cos^2(pix) is 1/2 x+1/(4pi)sin(2pix)+C

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integral of cos^2(pix)

\int cos^{2} (π x) d x

\frac{1}{2} x + \frac{1}{4 π} sin (2 π x) + C

Solution steps

\int cos^{2} (π x) d x

Rewrite using trig identities

= \int 1 - sin^{2} (π 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

= \int 1 d x - \int sin^{2} (π x) d x

\int 1 d x = x

\int sin^{2} (π x) d x = \frac{1}{2} (x - \frac{1}{2 π} sin (2 x π))

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

Simplify

x - \frac{1}{2} (x - \frac{1}{2 π} sin (2 x π)) : \frac{1}{2} x + \frac{1}{4 π} sin (2 π x)

= \frac{1}{2} x + \frac{1}{4 π} sin (2 π x)

Add a constant to the solution

= \frac{1}{2} x + \frac{1}{4 π} sin (2 π x) + C

\frac{2 - x e ^{x}}{x + e ^{x}}

2 y^{^{''}} + 27 y = 0, y (0) = 6, y^{^{'}} (0) = 5

\frac{d y}{d x} = (1 - y) cos (x)

\frac{d}{d x} (\frac{e ^{3 x}}{1 - x ^{2}})

\int \frac{28 x ^{3} - 56 x ^{2} + 3}{x ^{2} - 2 x} d x

What is the integral of cos^2(pix) ?
The integral of cos^2(pix) is 1/2 x+1/(4pi)sin(2pix)+C