What is the integral of xsin((npix)/3) ?

The integral of xsin((npix)/3) is 1/(pi^2n^2)(-3pinxcos((pinx)/3)+9sin((pinx)/3))+C

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integral of xsin((npix)/3)

\int x sin (\frac{nπ x}{3}) d x

\frac{1}{π ^{2} n ^{2}} (- 3 πn x cos (\frac{πn x}{3}) + 9 sin (\frac{πn x}{3})) + C

Solution steps

\int x sin (\frac{nπ x}{3}) d x

Apply u-substitution

: \int \frac{u sin ( \frac{u}{3} )}{π ^{2} n ^{2}} d u

= \int \frac{u sin ( \frac{u}{3} )}{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}{3}) d u

Apply Integration By Parts

: - 3 u cos (\frac{u}{3}) - \int - 3 cos (\frac{u}{3}) d u

= \frac{1}{π ^{2} n ^{2}} (- 3 u cos (\frac{u}{3}) - \int - 3 cos (\frac{u}{3}) d u)

\int - 3 cos (\frac{u}{3}) d u = - 9 sin (\frac{u}{3})

= \frac{1}{π ^{2} n ^{2}} (- 3 u cos (\frac{u}{3}) - (- 9 sin (\frac{u}{3})))

Substitute back

u = nπ x

= \frac{1}{π ^{2} n ^{2}} (- 3 nπ x cos (\frac{nπ x}{3}) - (- 9 sin (\frac{nπ x}{3})))

Simplify

= \frac{1}{π ^{2} n ^{2}} (- 3 πn x cos (\frac{πn x}{3}) + 9 sin (\frac{πn x}{3}))

Add a constant to the solution

= \frac{1}{π ^{2} n ^{2}} (- 3 πn x cos (\frac{πn x}{3}) + 9 sin (\frac{πn x}{3})) + C

\frac{1}{s ^{2} ( s ^{2} + 1 ^{2} )}

\int x \cdot arctan (x) d x

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

\frac{d}{d x} (\frac{2}{e ^{\frac{x}{2}} + e ^{- \frac{x}{2}}})

What is the integral of xsin((npix)/3) ?
The integral of xsin((npix)/3) is 1/(pi^2n^2)(-3pinxcos((pinx)/3)+9sin((pinx)/3))+C