What is the integral of (x-1)sin(nx) ?

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

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integral of (x-1)sin(nx)

\int (x - 1) sin (n x) d x

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

Solution steps

\int (x - 1) sin (n x) d x

Expand

(x - 1) sin (n x) : x sin (n x) - sin (n x)

= \int x sin (n x) - sin (n x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int x sin (n x) d x - \int sin (n x) d x

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

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

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

Simplify

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

Add a constant to the solution

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

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