What is the integral of xcos(pix) ?

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

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integral of xcos(pix)

\int x cos (π x) d x

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

Solution steps

\int x cos (π x) d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

\int (3 + e^{x} (y^{2} + 3 y + 2)) d x

\frac{\partial}{\partial x} (\frac{y}{( x + z )})

\int 4 x^{2} + 5 x - 3 d x

y = \frac{10}{1 + e ^{- x}}, (0, 5)

(3 x^{3} + 3) (x^{4} - 3 x)

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