What is the integral of sin^2((npix)/L) ?

The integral of sin^2((npix)/L) is 1/2 (x-L/(2pin)sin((2pinx)/L))+C

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integral of sin^2((npix)/L)

\int sin^{2} (\frac{nπ x}{L}) d x

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

Solution steps

\int sin^{2} (\frac{nπ x}{L}) d x

Rewrite using trig identities

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

\int 1 d x = x

\int cos (\frac{2 πn x}{L}) d x = \frac{L}{2 πn} sin (\frac{2 πn x}{L})

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

Add a constant to the solution

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

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What is the integral of sin^2((npix)/L) ?
The integral of sin^2((npix)/L) is 1/2 (x-L/(2pin)sin((2pinx)/L))+C