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

The integral of xcos((mpix)/2) is 2/(pim)xsin((pimx)/2)+4/(pi^2m^2)cos((pimx)/2)+C

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integral of xcos((mpix)/2)

\int x cos (\frac{mπ x}{2}) d x

\frac{2}{πm} x sin (\frac{πm x}{2}) + \frac{4}{π ^{2} m ^{2}} cos (\frac{πm x}{2}) + C

Solution steps

\int x cos (\frac{mπ x}{2}) d x

Apply Integration By Parts

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

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

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

= \frac{2}{πm} x sin (\frac{πm x}{2}) - (- \frac{4}{π ^{2} m ^{2}} cos (\frac{πm x}{2}))

Simplify

= \frac{2}{πm} x sin (\frac{πm x}{2}) + \frac{4}{π ^{2} m ^{2}} cos (\frac{πm x}{2})

Add a constant to the solution

= \frac{2}{πm} x sin (\frac{πm x}{2}) + \frac{4}{π ^{2} m ^{2}} cos (\frac{πm x}{2}) + C

\frac{d}{d x} ((\frac{3}{5})^{x})

\int sec (t) (4 sec (t) + 7 tan (t)) d t

\frac{d}{d y} (y e^{x})

x \to - \infty lim (x^{- 2})

\frac{d}{d x} (\frac{8 x}{9 - cot ( x )})

What is the integral of xcos((mpix)/2) ?
The integral of xcos((mpix)/2) is 2/(pim)xsin((pimx)/2)+4/(pi^2m^2)cos((pimx)/2)+C