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

The integral of e^{9x}sin(x) is -(e^{9x}cos(x))/(82)+(9e^{9x}sin(x))/(82)+C

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

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

- \frac{e ^{9 x} cos ( x )}{82} + \frac{9 e ^{9 x} sin ( x )}{82} + C

Solution steps

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

Apply Integration By Parts

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

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

Take the constant out:

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

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

Apply Integration By Parts

: e^{9 x} sin (x) - \int e^{9 x} \cdot 9 sin (x) d x

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

Take the constant out:

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

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

Therefore

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

Isolate

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

= - \frac{e ^{9 x} cos ( x )}{82} + \frac{9 e ^{9 x} sin ( x )}{82}

Add a constant to the solution

= - \frac{e ^{9 x} cos ( x )}{82} + \frac{9 e ^{9 x} sin ( x )}{82} + C

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What is the integral of e^{9x}sin(x) ?
The integral of e^{9x}sin(x) is -(e^{9x}cos(x))/(82)+(9e^{9x}sin(x))/(82)+C