What is the integral of xcos((npix)/(20)) ?

The integral of xcos((npix)/(20)) is 1/(pi^2n^2)(20pinxsin((pinx)/(20))+400cos((pinx)/(20)))+C

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integral of xcos((npix)/(20))

\int x cos (\frac{nπ x}{20}) d x

\frac{1}{π ^{2} n ^{2}} (20 πn x sin (\frac{πn x}{20}) + 400 cos (\frac{πn x}{20})) + C

Solution steps

\int x cos (\frac{nπ x}{20}) d x

Apply u-substitution

= \int \frac{u cos ( \frac{u}{20} )}{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 (\frac{u}{20}) d u

Apply Integration By Parts

= \frac{1}{π ^{2} n ^{2}} (20 u sin (\frac{u}{20}) - \int 20 sin (\frac{u}{20}) d u)

\int 20 sin (\frac{u}{20}) d u = - 400 cos (\frac{u}{20})

= \frac{1}{π ^{2} n ^{2}} (20 u sin (\frac{u}{20}) - (- 400 cos (\frac{u}{20})))

Substitute back

u = nπ x

= \frac{1}{π ^{2} n ^{2}} (20 nπ x sin (\frac{nπ x}{20}) - (- 400 cos (\frac{nπ x}{20})))

Simplify

= \frac{1}{π ^{2} n ^{2}} (20 πn x sin (\frac{πn x}{20}) + 400 cos (\frac{πn x}{20}))

Add a constant to the solution

= \frac{1}{π ^{2} n ^{2}} (20 πn x sin (\frac{πn x}{20}) + 400 cos (\frac{πn x}{20})) + C

What is the integral of xcos((npix)/(20)) ?
The integral of xcos((npix)/(20)) is 1/(pi^2n^2)(20pinxsin((pinx)/(20))+400cos((pinx)/(20)))+C