What is the integral of xsin((npix)/(10)) ?

The integral of xsin((npix)/(10)) is 1/(pi^2n^2)(-10pinxcos((pinx)/(10))+100sin((pinx)/(10)))+C

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integral of xsin((npix)/(10))

\int x sin (\frac{nπ x}{10}) d x

\frac{1}{π ^{2} n ^{2}} (- 10 πn x cos (\frac{πn x}{10}) + 100 sin (\frac{πn x}{10})) + C

Solution steps

\int x sin (\frac{nπ x}{10}) d x

Apply u-substitution

= \int \frac{u sin ( \frac{u}{10} )}{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}{10}) d u

Apply Integration By Parts

= \frac{1}{π ^{2} n ^{2}} (- 10 u cos (\frac{u}{10}) - \int - 10 cos (\frac{u}{10}) d u)

\int - 10 cos (\frac{u}{10}) d u = - 100 sin (\frac{u}{10})

= \frac{1}{π ^{2} n ^{2}} (- 10 u cos (\frac{u}{10}) - (- 100 sin (\frac{u}{10})))

Substitute back

u = nπ x

= \frac{1}{π ^{2} n ^{2}} (- 10 nπ x cos (\frac{nπ x}{10}) - (- 100 sin (\frac{nπ x}{10})))

Simplify

= \frac{1}{π ^{2} n ^{2}} (- 10 πn x cos (\frac{πn x}{10}) + 100 sin (\frac{πn x}{10}))

Add a constant to the solution

= \frac{1}{π ^{2} n ^{2}} (- 10 πn x cos (\frac{πn x}{10}) + 100 sin (\frac{πn x}{10})) + C

What is the integral of xsin((npix)/(10)) ?
The integral of xsin((npix)/(10)) is 1/(pi^2n^2)(-10pinxcos((pinx)/(10))+100sin((pinx)/(10)))+C