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

The integral of xsin((npix)/(20)) is 1/(pi^2n^2)(-20pinxcos((pinx)/(20))+400sin((pinx)/(20)))+C

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integral of xsin((npix)/(20))

\int x sin (\frac{nπ x}{20}) d x

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

Solution steps

\int x sin (\frac{nπ x}{20}) d x

Apply u-substitution

: \int \frac{u sin ( \frac{u}{20} )}{π ^{2} n ^{2}} d u

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

Apply Integration By Parts

: - 20 u cos (\frac{u}{20}) - \int - 20 cos (\frac{u}{20}) d u

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

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

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

Substitute back

u = nπ x

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

Simplify

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

Add a constant to the solution

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

x \to 2 - lim (\frac{x ^{2} - 4}{x - 2})

\frac{d}{d x} (\frac{4 x ( 2 x ^{2} + 3 )}{( 1 - 2 x ^{2} ) ^{3}})

\int \frac{3 x + 2}{( x + 5 ) ( x - 2 )} d x

s \to - 8 lim (4 (s) - 2)

\frac{d}{d x} (\frac{2}{x ^{2}} - 5 x)

What is the integral of xsin((npix)/(20)) ?
The integral of xsin((npix)/(20)) is 1/(pi^2n^2)(-20pinxcos((pinx)/(20))+400sin((pinx)/(20)))+C