What is the integral of e^{4x}sin(2x) ?

The integral of e^{4x}sin(2x) is -(e^{4x}cos(2x)}{10}+\frac{e^{4x}sin(2x))/5+C

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integral of e^{4x}sin(2x)

\int e^{4 x} sin (2 x) d x

- \frac{e ^{4 x} cos ( 2 x )}{10} + \frac{e ^{4 x} sin ( 2 x )}{5} + C

Solution steps

\int e^{4 x} sin (2 x) d x

Apply Integration By Parts

: - \frac{1}{2} e^{4 x} cos (2 x) - \int - 2 e^{4 x} cos (2 x) d x

= - \frac{1}{2} e^{4 x} cos (2 x) - \int - 2 e^{4 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^{4 x} cos (2 x) - (- 2 \cdot \int e^{4 x} cos (2 x) d x)

Apply Integration By Parts

: \frac{1}{2} e^{4 x} sin (2 x) - \int 2 e^{4 x} sin (2 x) d x

= - \frac{1}{2} e^{4 x} cos (2 x) - (- 2 (\frac{1}{2} e^{4 x} sin (2 x) - \int 2 e^{4 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^{4 x} cos (2 x) - (- 2 (\frac{1}{2} e^{4 x} sin (2 x) - 2 \cdot \int e^{4 x} sin (2 x) d x))

Therefore

\int e^{4 x} sin (2 x) d x = - \frac{1}{2} e^{4 x} cos (2 x) - (- 2 (\frac{1}{2} e^{4 x} sin (2 x) - 2 \cdot \int e^{4 x} sin (2 x) d x))

Isolate

\int e^{4 x} sin (2 x) d x

= - \frac{e ^{4 x} cos ( 2 x )}{10} + \frac{e ^{4 x} sin ( 2 x )}{5}

Add a constant to the solution

= - \frac{e ^{4 x} cos ( 2 x )}{10} + \frac{e ^{4 x} sin ( 2 x )}{5} + C

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What is the integral of e^{4x}sin(2x) ?
The integral of e^{4x}sin(2x) is -(e^{4x}cos(2x)}{10}+\frac{e^{4x}sin(2x))/5+C