What is the integral of tcos(nt) ?

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

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integral of tcos(nt)

\int t 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

= \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

: u sin (u) - \int sin (u) d u

= \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

k \to \infty lim (\frac{k ^{\frac{1}{k}}}{ln ( k )})

\frac{d}{d s} (\frac{s ^{2} - 4 s + 20}{( s + 2 ) ( s + 4 )})

\int e^{t a n (6 x)} sec^{2} (6 x) d x

y^{^{'}} = 6 x^{2} y

\frac{d y}{d x}, 4 x = y^{2}

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