What is the integral of x*cos(npix) ?

The integral of x*cos(npix) is 1/(pi^2n^2)(pinxsin(pinx)+cos(pinx))+C

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integral of x*cos(npix)

\int x \cdot cos (nπ x) d x

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

Solution steps

\int x cos (nπ x) d x

Apply u-substitution

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

Apply Integration By Parts

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

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

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

Substitute back

u = n x π

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

Simplify

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

Add a constant to the solution

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

What is the integral of x*cos(npix) ?
The integral of x*cos(npix) is 1/(pi^2n^2)(pinxsin(pinx)+cos(pinx))+C