What is the integral of xsin(pix) ?

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

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integral of xsin(pix)

\int x sin (π x) d x

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

Solution steps

\int x sin (π x) d x

Apply Integration By Parts

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

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

Simplify

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

Add a constant to the solution

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

\frac{\partial}{\partial x} (k ln (x^{2} + y^{2}))

n = 1 \sum \infty (- \frac{1}{2})^{n + 1}

\int cos^{5} (x) tan^{4} (x) d x

\frac{d y}{d x} (1 + e^{x}) = e^{x - y}

f (x) = e^{x}, at x = ln (2)

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