What is the integral of e^{6x}sin(6x) ?

The integral of e^{6x}sin(6x) is -(e^{6x}cos(6x))/(12)+(e^{6x}sin(6x))/(12)+C

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integral of e^{6x}sin(6x)

\int e^{6 x} sin (6 x) d x

- \frac{e ^{6 x} cos ( 6 x )}{12} + \frac{e ^{6 x} sin ( 6 x )}{12} + C

Solution steps

\int e^{6 x} sin (6 x) d x

Apply Integration By Parts

= - \frac{1}{6} e^{6 x} cos (6 x) - \int - e^{6 x} cos (6 x) d x

Take the constant out:

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

= - \frac{1}{6} e^{6 x} cos (6 x) - (- \int e^{6 x} cos (6 x) d x)

Apply Integration By Parts

= - \frac{1}{6} e^{6 x} cos (6 x) - (- (\frac{1}{6} e^{6 x} sin (6 x) - \int e^{6 x} sin (6 x) d x))

Therefore

\int e^{6 x} sin (6 x) d x = - \frac{1}{6} e^{6 x} cos (6 x) - (- (\frac{1}{6} e^{6 x} sin (6 x) - \int e^{6 x} sin (6 x) d x))

Isolate

\int e^{6 x} sin (6 x) d x

= - \frac{e ^{6 x} cos ( 6 x )}{12} + \frac{e ^{6 x} sin ( 6 x )}{12}

Add a constant to the solution

= - \frac{e ^{6 x} cos ( 6 x )}{12} + \frac{e ^{6 x} sin ( 6 x )}{12} + C

What is the integral of e^{6x}sin(6x) ?
The integral of e^{6x}sin(6x) is -(e^{6x}cos(6x))/(12)+(e^{6x}sin(6x))/(12)+C