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

The integral of cos((2pinx)/L) is L/(2pin)sin((2pinx)/L)+C

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integral of cos((2pinx)/L)

\int cos (\frac{2 πn x}{L}) d x

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

Solution steps

\int cos (\frac{2 πn x}{L}) d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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What is the integral of cos((2pinx)/L) ?
The integral of cos((2pinx)/L) is L/(2pin)sin((2pinx)/L)+C