What is the integral of e^{9x}sin(2x) ?

The integral of e^{9x}sin(2x) is -(2e^{9x}cos(2x))/(85)+(9e^{9x}sin(2x))/(85)+C

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integral of e^{9x}sin(2x)

\int e^{9 x} sin (2 x) d x

- \frac{2 e ^{9 x} cos ( 2 x )}{85} + \frac{9 e ^{9 x} sin ( 2 x )}{85} + C

Solution steps

\int e^{9 x} sin (2 x) d x

Apply Integration By Parts

: - \frac{1}{2} e^{9 x} cos (2 x) - \int - \frac{9}{2} e^{9 x} cos (2 x) d x

= - \frac{1}{2} e^{9 x} cos (2 x) - \int - \frac{9}{2} e^{9 x} cos (2 x) d x

Take the constant out:

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

= - \frac{1}{2} e^{9 x} cos (2 x) - (- \frac{9}{2} \cdot \int e^{9 x} cos (2 x) d x)

Apply Integration By Parts

: \frac{1}{2} e^{9 x} sin (2 x) - \int \frac{9}{2} e^{9 x} sin (2 x) d x

= - \frac{1}{2} e^{9 x} cos (2 x) - (- \frac{9}{2} (\frac{1}{2} e^{9 x} sin (2 x) - \int \frac{9}{2} e^{9 x} sin (2 x) d x))

Take the constant out:

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

= - \frac{1}{2} e^{9 x} cos (2 x) - (- \frac{9}{2} (\frac{1}{2} e^{9 x} sin (2 x) - \frac{9}{2} \cdot \int e^{9 x} sin (2 x) d x))

Therefore

\int e^{9 x} sin (2 x) d x = - \frac{1}{2} e^{9 x} cos (2 x) - (- \frac{9}{2} (\frac{1}{2} e^{9 x} sin (2 x) - \frac{9}{2} \cdot \int e^{9 x} sin (2 x) d x))

Isolate

\int e^{9 x} sin (2 x) d x

= - \frac{2 e ^{9 x} cos ( 2 x )}{85} + \frac{9 e ^{9 x} sin ( 2 x )}{85}

Add a constant to the solution

= - \frac{2 e ^{9 x} cos ( 2 x )}{85} + \frac{9 e ^{9 x} sin ( 2 x )}{85} + C

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What is the integral of e^{9x}sin(2x) ?
The integral of e^{9x}sin(2x) is -(2e^{9x}cos(2x))/(85)+(9e^{9x}sin(2x))/(85)+C