What is the integral of xsin((npix)/5) ?

The integral of xsin((npix)/5) is 1/(pi^2n^2)(-5pinxcos((pinx)/5)+25sin((pinx)/5))+C

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integral of xsin((npix)/5)

\int x sin (\frac{nπ x}{5}) d x

\frac{1}{π ^{2} n ^{2}} (- 5 πn x cos (\frac{πn x}{5}) + 25 sin (\frac{πn x}{5})) + C

Solution steps

\int x sin (\frac{nπ x}{5}) d x

Apply u-substitution

= \int \frac{u sin ( \frac{u}{5} )}{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}{5}) d u

Apply Integration By Parts

= \frac{1}{π ^{2} n ^{2}} (- 5 u cos (\frac{u}{5}) - \int - 5 cos (\frac{u}{5}) d u)

\int - 5 cos (\frac{u}{5}) d u = - 25 sin (\frac{u}{5})

= \frac{1}{π ^{2} n ^{2}} (- 5 u cos (\frac{u}{5}) - (- 25 sin (\frac{u}{5})))

Substitute back

u = nπ x

= \frac{1}{π ^{2} n ^{2}} (- 5 nπ x cos (\frac{nπ x}{5}) - (- 25 sin (\frac{nπ x}{5})))

Simplify

= \frac{1}{π ^{2} n ^{2}} (- 5 πn x cos (\frac{πn x}{5}) + 25 sin (\frac{πn x}{5}))

Add a constant to the solution

= \frac{1}{π ^{2} n ^{2}} (- 5 πn x cos (\frac{πn x}{5}) + 25 sin (\frac{πn x}{5})) + C

What is the integral of xsin((npix)/5) ?
The integral of xsin((npix)/5) is 1/(pi^2n^2)(-5pinxcos((pinx)/5)+25sin((pinx)/5))+C