What is the integral of e^{3t}sin(t) ?

The integral of e^{3t}sin(t) is -(e^{3t}cos(t))/(10)+(3e^{3t}sin(t))/(10)+C

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integral of e^{3t}sin(t)

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

- \frac{e ^{3 t} cos ( t )}{10} + \frac{3 e ^{3 t} sin ( t )}{10} + C

Solution steps

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

Apply Integration By Parts

: - e^{3 t} cos (t) - \int - 3 e^{3 t} cos (t) d t

= - e^{3 t} cos (t) - \int - 3 e^{3 t} cos (t) d t

Take the constant out:

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

= - e^{3 t} cos (t) - (- 3 \cdot \int e^{3 t} cos (t) d t)

Apply Integration By Parts

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

= - e^{3 t} cos (t) - (- 3 (e^{3 t} sin (t) - \int e^{3 t} \cdot 3 sin (t) d t))

Take the constant out:

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

= - e^{3 t} cos (t) - (- 3 (e^{3 t} sin (t) - 3 \cdot \int e^{3 t} sin (t) d t))

Therefore

\int e^{3 t} sin (t) d t = - e^{3 t} cos (t) - (- 3 (e^{3 t} sin (t) - 3 \cdot \int e^{3 t} sin (t) d t))

Isolate

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

= - \frac{e ^{3 t} cos ( t )}{10} + \frac{3 e ^{3 t} sin ( t )}{10}

Add a constant to the solution

= - \frac{e ^{3 t} cos ( t )}{10} + \frac{3 e ^{3 t} sin ( t )}{10} + C

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What is the integral of e^{3t}sin(t) ?
The integral of e^{3t}sin(t) is -(e^{3t}cos(t))/(10)+(3e^{3t}sin(t))/(10)+C