What is the integral of tsin(at) ?

The integral of tsin(at) is -1/a tcos(at)+1/(a^2)sin(at)+C

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integral of tsin(at)

\int t sin (a t)

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

Solution steps

\int t sin (a t) d t

Apply Integration By Parts

: - \frac{1}{a} t cos (a t) - \int - \frac{1}{a} cos (a t) d t

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

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

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

Simplify

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

Add a constant to the solution

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

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What is the integral of tsin(at) ?
The integral of tsin(at) is -1/a tcos(at)+1/(a^2)sin(at)+C