What is the integral of 2xcos(npix) ?

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

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integral of 2xcos(npix)

\int 2 x cos (nπ x) d x

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

Solution steps

\int 2 x cos (nπ x) d x

Take the constant out:

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

= 2 \cdot \int x cos (nπ x) d x

Apply u-substitution

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

= 2 \cdot \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

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

Apply Integration By Parts

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

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

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

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

Substitute back

u = n x π

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

Simplify

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

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

Add a constant to the solution

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

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