What is the integral of (x-1)sin(pix) ?

The integral of (x-1)sin(pix) is -1/pi cos(pix)(x-1)+1/(pi^2)sin(pix)+C

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integral of (x-1)sin(pix)

\int (x - 1) sin (π x) d x

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

Solution steps

\int (x - 1) sin (π x) d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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What is the integral of (x-1)sin(pix) ?
The integral of (x-1)sin(pix) is -1/pi cos(pix)(x-1)+1/(pi^2)sin(pix)+C