What is the integral of (x-4)sin(pix) ?

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

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integral of (x-4)sin(pix)

\int (x - 4) sin (π x) d x

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

Solution steps

\int (x - 4) sin (π x) d x

Apply Integration By Parts

: - \frac{1}{π} cos (π x) (x - 4) - \int - \frac{1}{π} cos (π x) d x

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

Simplify

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

Add a constant to the solution

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

\int \frac{4 x}{( 1 + x ^{2} )} d x

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\frac{\partial}{\partial y} (x (e^{y} + e^{- y}))

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\int sin^{3} (d) x d x

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