What is the integral of xsin(2pix) ?

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

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integral of xsin(2pix)

\int x sin (2 π x) d x

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

Solution steps

\int x sin (2 π x) d x

Apply Integration By Parts

: - \frac{1}{2 π} x cos (2 π x) - \int - \frac{1}{2 π} cos (2 π x) d x

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

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

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

Simplify

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

Add a constant to the solution

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

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