What is the integral of cos^2((3pix)/L) ?

The integral of cos^2((3pix)/L) is 1/2 (x+1/(6pi)Lsin((6pix)/L))+C

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integral of cos^2((3pix)/L)

\int cos^{2} (\frac{3 π x}{L}) d x

\frac{1}{2} (x + \frac{1}{6 π} L sin (\frac{6 π x}{L})) + C

Solution steps

\int cos^{2} (\frac{3 π x}{L}) d x

Rewrite using trig identities

= \int \frac{1}{2} (1 + cos (\frac{6 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{6 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{6 π x}{L}) d x)

\int 1 d x = x

\int cos (\frac{6 π x}{L}) d x = \frac{1}{6 π} L sin (\frac{6 π x}{L})

= \frac{1}{2} (x + \frac{1}{6 π} L sin (\frac{6 π x}{L}))

Add a constant to the solution

= \frac{1}{2} (x + \frac{1}{6 π} L sin (\frac{6 π x}{L})) + C

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\frac{\partial}{\partial x} ((x y + yz + x z)^{2})

What is the integral of cos^2((3pix)/L) ?
The integral of cos^2((3pix)/L) is 1/2 (x+1/(6pi)Lsin((6pix)/L))+C