What is the integral of xcos((nx)/2) ?

The integral of xcos((nx)/2) is 2/n xsin((nx)/2)+4/(n^2)cos((nx)/2)+C

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integral of xcos((nx)/2)

\int x cos (\frac{n x}{2}) d x

\frac{2}{n} x sin (\frac{n x}{2}) + \frac{4}{n ^{2}} cos (\frac{n x}{2}) + C

Solution steps

\int x cos (\frac{n x}{2}) d x

Apply Integration By Parts

: \frac{2}{n} x sin (\frac{n x}{2}) - \int \frac{2}{n} sin (\frac{n x}{2}) d x

= \frac{2}{n} x sin (\frac{n x}{2}) - \int \frac{2}{n} sin (\frac{n x}{2}) d x

\int \frac{2}{n} sin (\frac{n x}{2}) d x = - \frac{4}{n ^{2}} cos (\frac{n x}{2})

= \frac{2}{n} x sin (\frac{n x}{2}) - (- \frac{4}{n ^{2}} cos (\frac{n x}{2}))

Simplify

= \frac{2}{n} x sin (\frac{n x}{2}) + \frac{4}{n ^{2}} cos (\frac{n x}{2})

Add a constant to the solution

= \frac{2}{n} x sin (\frac{n x}{2}) + \frac{4}{n ^{2}} cos (\frac{n x}{2}) + C

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What is the integral of xcos((nx)/2) ?
The integral of xcos((nx)/2) is 2/n xsin((nx)/2)+4/(n^2)cos((nx)/2)+C