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

The integral of cos((2pix)/L) is

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integral of cos((2pix)/L)

\int cos (\frac{2 π x}{L}) d x

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

Solution steps

\int cos (\frac{2 π x}{L}) d x

Apply Integration By Parts

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

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

= x cos (\frac{2 π x}{L}) - (- \frac{π}{L} x^{2} sin (\frac{2 π x}{L}) + \frac{π}{L} x^{2} sin (\frac{2 π x - L}{L} + 1) + x cos (\frac{2 π x - L}{L} + 1) - \frac{L}{2 π} sin (\frac{2 π x - L}{L} + 1))

Simplify

= x cos (\frac{2 π x}{L}) + \frac{π}{L} x^{2} sin (\frac{2 π x}{L}) - \frac{π}{L} x^{2} sin (\frac{2 π x - L}{L} + 1) - x cos (\frac{2 π x - L}{L} + 1) + \frac{L}{2 π} sin (\frac{2 π x - L}{L} + 1)

Add a constant to the solution

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