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

The integral of t*cos(nt) is 1/(n^2)(ntsin(nt)+cos(nt))+C

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integral of t*cos(nt)

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

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

Solution steps

\int t cos (n t) d t

Apply u-substitution

= \int \frac{u cos ( u )}{n ^{2}} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{n ^{2}} \cdot \int u cos (u) d u

Apply Integration By Parts

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

\int sin (u) d u = - cos (u)

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

Substitute back

u = n t

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

Simplify

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

Add a constant to the solution

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

What is the integral of t*cos(nt) ?
The integral of t*cos(nt) is 1/(n^2)(ntsin(nt)+cos(nt))+C