What is the integral of tcos(at) ?

The integral of tcos(at) is 1/(a^2)(atsin(at)+cos(at))+C

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integral of tcos(at)

\int t cos (a t) d t

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

Solution steps

\int t cos (a t) d t

Apply u-substitution

: \int \frac{u cos ( u )}{a ^{2}} d u

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

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

Substitute back

u = a t

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

Simplify

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

Add a constant to the solution

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

\int tan^{4} (x se) c^{6} x d x

\frac{d}{d x} (9 e^{1 - 3 x})

y^{^{''}} + 4 y^{^{'}} + 4 y = t

\int \frac{1}{3 + ( x - 2 ) ^{2}} d x

\frac{x ^{4}}{8} + \frac{1}{4 x ^{2}}

What is the integral of tcos(at) ?
The integral of tcos(at) is 1/(a^2)(atsin(at)+cos(at))+C