What is the integral of xsin(pikx) ?

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

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integral of xsin(pikx)

\int x sin (πk x) d x

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

Solution steps

\int x sin (πk x) d x

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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