What is the integral of xcos(n*pi/3*x) ?

The integral of xcos(n*pi/3*x) is 9/(pi^2n^2)(pi/3 nxsin(pi/3 nx)+cos(pi/3 nx))+C

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integral of xcos(npi/3x)

\int x cos (n \cdot \frac{π}{3} \cdot x) d x

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

Solution steps

\int x cos (n \frac{π}{3} x) d x

Apply u-substitution

: \int \frac{9 u cos ( u )}{π ^{2} n ^{2}} d u

= \int \frac{9 u cos ( u )}{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{9}{π ^{2} n ^{2}} \cdot \int u cos (u) d u

Apply Integration By Parts

: u sin (u) - \int sin (u) d u

= \frac{9}{π ^{2} n ^{2}} (u sin (u) - \int sin (u) d u)

\int sin (u) d u = - cos (u)

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

Substitute back

u = n \frac{π}{3} x

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

Simplify

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

Add a constant to the solution

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

x \to - \infty lim (x \cdot (x))

y = x^{2}, x = 3, x = - 3

f (x) = - x^{2} + 5

y = sin (3 ln (x))

\int \frac{1}{7 x - 3} d x

What is the integral of xcos(n*pi/3*x) ?
The integral of xcos(n*pi/3*x) is 9/(pi^2n^2)(pi/3 nxsin(pi/3 nx)+cos(pi/3 nx))+C