What is the integral of xcos(npix) ?

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

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integral of xcos(npix)

\int x 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

= \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

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

= \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

x^{3} - 25 x + 5

\frac{d}{d x} (2 x^{3} - 11.7 x^{2} + 17.7 x - 5)

(1 + x)^{\frac{1}{2}}

\frac{d}{d x} ((x^{\frac{1}{2}} + y^{\frac{1}{2}})^{2})

(2 x^{3} - y) d x - (x + 2 y) d y = 0

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