What is the integral of e^xsin(3x) ?

The integral of e^xsin(3x) is -(3e^xcos(3x))/(10)+(e^xsin(3x))/(10)+C

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integral of e^xsin(3x)

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

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

Solution steps

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

Apply Integration By Parts

= - \frac{1}{3} e^{x} cos (3 x) - \int - \frac{1}{3} e^{x} cos (3 x) d x

Take the constant out:

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

= - \frac{1}{3} e^{x} cos (3 x) - (- \frac{1}{3} \cdot \int e^{x} cos (3 x) d x)

Apply Integration By Parts

= - \frac{1}{3} e^{x} cos (3 x) - (- \frac{1}{3} (e^{x} \frac{1}{3} sin (3 x) - \int e^{x} \frac{1}{3} sin (3 x) d x))

Take the constant out:

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

= - \frac{1}{3} e^{x} cos (3 x) - (- \frac{1}{3} (e^{x} \frac{1}{3} sin (3 x) - \frac{1}{3} \cdot \int e^{x} sin (3 x) d x))

Therefore

\int e^{x} sin (3 x) d x = - \frac{1}{3} e^{x} cos (3 x) - (- \frac{1}{3} (e^{x} \frac{1}{3} sin (3 x) - \frac{1}{3} \cdot \int e^{x} sin (3 x) d x))

Isolate

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

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

Add a constant to the solution

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

What is the integral of e^xsin(3x) ?
The integral of e^xsin(3x) is -(3e^xcos(3x))/(10)+(e^xsin(3x))/(10)+C