What is the integral of xcos(2pix) ?

The integral of xcos(2pix) is 1/(2pi)xsin(2pix)+1/(4pi^2)cos(2pix)+C

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integral of xcos(2pix)

\int x cos (2 π x) d x

\frac{1}{2 π} x sin (2 π x) + \frac{1}{4 π ^{2}} cos (2 π x) + C

Solution steps

\int x cos (2 π x) d x

Apply Integration By Parts

= \frac{1}{2 π} x sin (2 π x) - \int \frac{1}{2 π} sin (2 π x) d x

\int \frac{1}{2 π} sin (2 π x) d x = - \frac{1}{4 π ^{2}} cos (2 π x)

= \frac{1}{2 π} x sin (2 π x) - (- \frac{1}{4 π ^{2}} cos (2 π x))

Simplify

= \frac{1}{2 π} x sin (2 π x) + \frac{1}{4 π ^{2}} cos (2 π x)

Add a constant to the solution

= \frac{1}{2 π} x sin (2 π x) + \frac{1}{4 π ^{2}} cos (2 π x) + C

What is the integral of xcos(2pix) ?
The integral of xcos(2pix) is 1/(2pi)xsin(2pix)+1/(4pi^2)cos(2pix)+C