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

The integral of xcos((npix)/3) is 1/(pi^2n^2)(3pinxsin((pinx)/3)+9cos((pinx)/3))+C

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integral of xcos((npix)/3)

\int x cos (\frac{nπ x}{3}) d x

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

Solution steps

\int x cos (\frac{nπ x}{3}) d x

Apply u-substitution

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

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

Apply Integration By Parts

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

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

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

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

Substitute back

u = nπ x

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

Simplify

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

Add a constant to the solution

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

\int sin (r) d r

y = 9 e^{x} + x

\frac{d y}{d x} = e^{2 x - 3 y}

\frac{\partial}{\partial x} (7 x cos (x) cos (y))

\frac{d y}{d t} = 8 - \frac{y}{200 + 3 t}

What is the integral of xcos((npix)/3) ?
The integral of xcos((npix)/3) is 1/(pi^2n^2)(3pinxsin((pinx)/3)+9cos((pinx)/3))+C