What is the integral of t*sin(nt) ?

The integral of t*sin(nt) is -1/n tcos(nt)+1/(n^2)sin(nt)+C

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integral of t*sin(nt)

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

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

Solution steps

\int t sin (n t) d t

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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What is the integral of t*sin(nt) ?
The integral of t*sin(nt) is -1/n tcos(nt)+1/(n^2)sin(nt)+C