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

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

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

\int sin^{2} (\frac{2 π x}{L}) d x

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

Solution steps

\int sin^{2} (\frac{2 π x}{L}) d x

Rewrite using trig identities

= \int \frac{1}{2} (1 - cos (\frac{4 π 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{4 π 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{4 π x}{L}) d x)

\int 1 d x = x

\int cos (\frac{4 π x}{L}) d x = \frac{1}{4 π} L sin (\frac{4 π x}{L})

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

Add a constant to the solution

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

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

x \to 3 lim (\frac{7 x ^{2} - 3 x}{x})

\frac{1}{4} (\frac{d y}{d t}) + 10 y = 5

\int sec^{2} (x t a) n^{2} x d x

f (x) = x^{3} - \frac{3}{2} x^{2} + x, at x = 1

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