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

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

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integral of cos((2pix)/a)

\int cos (\frac{2 π x}{a}) d x

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

Solution steps

\int cos (\frac{2 π x}{a}) d x

Apply Integration By Parts

: x cos (\frac{2 π x}{a}) - \int - \frac{2 π}{a} x sin (\frac{2 π x}{a}) d x

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

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

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

Simplify

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

Add a constant to the solution

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

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