What is the integral of xcos(pikx) ?

The integral of xcos(pikx) is 1/(pi^2k^2)(pikxsin(pikx)+cos(pikx))+C

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integral of xcos(pikx)

\int x cos (πk x) d x

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

Solution steps

\int x cos (πk x) d x

Apply u-substitution

: \int \frac{u cos ( u )}{π ^{2} k ^{2}} d u

= \int \frac{u cos ( u )}{π ^{2} k ^{2}} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{π ^{2} k ^{2}} \cdot \int u cos (u) d u

Apply Integration By Parts

: u sin (u) - \int sin (u) d u

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

\int sin (u) d u = - cos (u)

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

Substitute back

u = k x π

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

Simplify

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

Add a constant to the solution

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

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What is the integral of xcos(pikx) ?
The integral of xcos(pikx) is 1/(pi^2k^2)(pikxsin(pikx)+cos(pikx))+C