What is the integral of x*sin(nx) ?

The integral of x*sin(nx) is -1/n xcos(nx)+1/(n^2)sin(nx)+C

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integral of x*sin(nx)

\int x \cdot sin (n x) d x

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

Solution steps

\int x sin (n x) d x

Apply Integration By Parts

= - \frac{1}{n} x cos (n x) - \int - \frac{1}{n} cos (n x) d x

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

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

Simplify

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

Add a constant to the solution

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

What is the integral of x*sin(nx) ?
The integral of x*sin(nx) is -1/n xcos(nx)+1/(n^2)sin(nx)+C