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

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

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integral of x*sin(ax)

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

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

Solution steps

\int x sin (a x) d x

Apply Integration By Parts

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

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

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

Simplify

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

Add a constant to the solution

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

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