What is the integral of xsin((npix)/4) ?

The integral of xsin((npix)/4) is 1/(pi^2n^2)(-4pinxcos((pinx)/4)+16sin((pinx)/4))+C

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integral of xsin((npix)/4)

\int x sin (\frac{nπ x}{4}) d x

\frac{1}{π ^{2} n ^{2}} (- 4 πn x cos (\frac{πn x}{4}) + 16 sin (\frac{πn x}{4})) + C

Solution steps

\int x sin (\frac{nπ x}{4}) d x

Apply u-substitution

: \int \frac{u sin ( \frac{u}{4} )}{π ^{2} n ^{2}} d u

= \int \frac{u sin ( \frac{u}{4} )}{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}{4}) d u

Apply Integration By Parts

: - 4 u cos (\frac{u}{4}) - \int - 4 cos (\frac{u}{4}) d u

= \frac{1}{π ^{2} n ^{2}} (- 4 u cos (\frac{u}{4}) - \int - 4 cos (\frac{u}{4}) d u)

\int - 4 cos (\frac{u}{4}) d u = - 16 sin (\frac{u}{4})

= \frac{1}{π ^{2} n ^{2}} (- 4 u cos (\frac{u}{4}) - (- 16 sin (\frac{u}{4})))

Substitute back

u = nπ x

= \frac{1}{π ^{2} n ^{2}} (- 4 nπ x cos (\frac{nπ x}{4}) - (- 16 sin (\frac{nπ x}{4})))

Simplify

= \frac{1}{π ^{2} n ^{2}} (- 4 πn x cos (\frac{πn x}{4}) + 16 sin (\frac{πn x}{4}))

Add a constant to the solution

= \frac{1}{π ^{2} n ^{2}} (- 4 πn x cos (\frac{πn x}{4}) + 16 sin (\frac{πn x}{4})) + C

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What is the integral of xsin((npix)/4) ?
The integral of xsin((npix)/4) is 1/(pi^2n^2)(-4pinxcos((pinx)/4)+16sin((pinx)/4))+C