What is the integral of (pi-x)cos(pix) ?

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

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integral of (pi-x)cos(pix)

\int (π - x) cos (π x) d x

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

Solution steps

\int (π - x) cos (π x) d x

Expand

(π - x) cos (π x) : π cos (π x) - x cos (π x)

= \int π cos (π x) - x cos (π x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int π cos (π x) d x - \int x cos (π x) d x

\int π cos (π x) d x = sin (π x)

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

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

Simplify

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

Add a constant to the solution

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

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