What is the integral of e^{at}sin(t) ?

The integral of e^{at}sin(t) is -(e^{at}cos(t))/(1+a^2)+(ae^{at}sin(t))/(1+a^2)+C

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integral of e^{at}sin(t)

\int e^{a t} sin (t) d t

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

Solution steps

\int e^{a t} sin (t) d t

Apply Integration By Parts

: - e^{a t} cos (t) - \int - a e^{a t} cos (t) d t

= - e^{a t} cos (t) - \int - a e^{a t} cos (t) d t

Take the constant out:

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

= - e^{a t} cos (t) - (- a \cdot \int e^{a t} cos (t) d t)

Apply Integration By Parts

: e^{a t} sin (t) - \int e^{a t} a sin (t) d t

= - e^{a t} cos (t) - (- a (e^{a t} sin (t) - \int e^{a t} a sin (t) d t))

Take the constant out:

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

= - e^{a t} cos (t) - (- a (e^{a t} sin (t) - a \cdot \int e^{a t} sin (t) d t))

Therefore

\int e^{a t} sin (t) d t = - e^{a t} cos (t) - (- a (e^{a t} sin (t) - a \cdot \int e^{a t} sin (t) d t))

Isolate

\int e^{a t} sin (t) d t

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

Add a constant to the solution

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

\int sec^{2} (\frac{x}{4}) d x

\int e^{3 t} sin (t) d t

\frac{\partial}{\partial y} (\frac{2 y}{x})

\frac{1}{s ^{3} + 4 s ^{2} + 3 s}

\int (arctan (x)) d x

What is the integral of e^{at}sin(t) ?
The integral of e^{at}sin(t) is -(e^{at}cos(t))/(1+a^2)+(ae^{at}sin(t))/(1+a^2)+C