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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

\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

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