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

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

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integral of sin^2(pix)

\int sin^{2} (π x) d x

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

Solution steps

\int sin^{2} (π x) d x

Rewrite using trig identities

= \int \frac{1 - cos ( 2 π 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 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

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

\int 1 d x = x

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

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

Add a constant to the solution

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

sin (- 2 t)

\int_{- 1}^{7} (3 x - 9) d x

x \to 0 + lim (e^{x} - 1)

\frac{d}{d x} (∣ x + 2 ∣)

x \to 0 - lim (\frac{1}{x} e^{\frac{1}{x}})

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