What is the integral of xcos(xn) ?

The integral of xcos(xn) is 1/(n^2)(nxsin(nx)+cos(nx))+C

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integral of xcos(xn)

\int x cos (x n) d x

\frac{1}{n ^{2}} (n x sin (n x) + cos (n x)) + C

Solution steps

\int x cos (x n) d x

Apply u-substitution

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

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

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{n ^{2}} \cdot \int u cos (u) d u

Apply Integration By Parts

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

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

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

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

Substitute back

u = x n

= \frac{1}{n ^{2}} (x n sin (x n) - (- cos (x n)))

Simplify

= \frac{1}{n ^{2}} (n x sin (n x) + cos (n x))

Add a constant to the solution

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

\int x (6 - x) d x

\frac{s}{( s ^{2} - k ^{2} )}

\frac{\partial}{\partial x} ((x^{3}) x^{2} + 9)

x \to 1 - lim (\frac{x ^{2} - 1}{x - 1})

\int \frac{x}{( x + 1 ) ( x + 2 ) ^{2}} d x

What is the integral of xcos(xn) ?
The integral of xcos(xn) is 1/(n^2)(nxsin(nx)+cos(nx))+C