What is the integral of tcos(nwt) ?

The integral of tcos(nwt) is 1/(wn)tsin(wnt)+1/(w^2n^2)cos(wnt)+C

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integral of tcos(nwt)

\int t cos (n wt) d t

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

Solution steps

\int t cos (n wt) d t

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

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What is the integral of tcos(nwt) ?
The integral of tcos(nwt) is 1/(wn)tsin(wnt)+1/(w^2n^2)cos(wnt)+C