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

The integral of e^{6x}sin(x) is -(e^{6x}cos(x))/(37)+(6e^{6x}sin(x))/(37)+C

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

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

- \frac{e ^{6 x} cos ( x )}{37} + \frac{6 e ^{6 x} sin ( x )}{37} + C

Solution steps

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

Apply Integration By Parts

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

Take the constant out:

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

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

Apply Integration By Parts

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

Take the constant out:

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

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

Therefore

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

Isolate

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

= - \frac{e ^{6 x} cos ( x )}{37} + \frac{6 e ^{6 x} sin ( x )}{37}

Add a constant to the solution

= - \frac{e ^{6 x} cos ( x )}{37} + \frac{6 e ^{6 x} sin ( x )}{37} + C

What is the integral of e^{6x}sin(x) ?
The integral of e^{6x}sin(x) is -(e^{6x}cos(x))/(37)+(6e^{6x}sin(x))/(37)+C