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

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

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integral of e^{3x}*sin(x)

\int e^{3 x} \cdot sin (x) d x

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

Solution steps

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Therefore

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

Isolate

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

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

Add a constant to the solution

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

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