What is the integral of e^{2x+3}*sin(4x) ?

The integral of e^{2x+3}*sin(4x) is -(e^{2x+3}cos(4x))/5+(e^{2x+3}sin(4x))/(10)+C

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integral of e^{2x+3}*sin(4x)

\int e^{2 x + 3} \cdot sin (4 x) d x

- \frac{e ^{2 x + 3} cos ( 4 x )}{5} + \frac{e ^{2 x + 3} sin ( 4 x )}{10} + C

Solution steps

\int e^{2 x + 3} sin (4 x) d x

Apply Integration By Parts

: - \frac{1}{4} e^{2 x + 3} cos (4 x) - \int - \frac{1}{2} e^{2 x + 3} cos (4 x) d x

= - \frac{1}{4} e^{2 x + 3} cos (4 x) - \int - \frac{1}{2} e^{2 x + 3} cos (4 x) d x

Take the constant out:

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

= - \frac{1}{4} e^{2 x + 3} cos (4 x) - (- \frac{1}{2} \cdot \int e^{2 x + 3} cos (4 x) d x)

Apply Integration By Parts

: \frac{1}{4} e^{2 x + 3} sin (4 x) - \int \frac{1}{2} e^{2 x + 3} sin (4 x) d x

= - \frac{1}{4} e^{2 x + 3} cos (4 x) - (- \frac{1}{2} (\frac{1}{4} e^{2 x + 3} sin (4 x) - \int \frac{1}{2} e^{2 x + 3} sin (4 x) d x))

Take the constant out:

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

= - \frac{1}{4} e^{2 x + 3} cos (4 x) - (- \frac{1}{2} (\frac{1}{4} e^{2 x + 3} sin (4 x) - \frac{1}{2} \cdot \int e^{2 x + 3} sin (4 x) d x))

Therefore

\int e^{2 x + 3} sin (4 x) d x = - \frac{1}{4} e^{2 x + 3} cos (4 x) - (- \frac{1}{2} (\frac{1}{4} e^{2 x + 3} sin (4 x) - \frac{1}{2} \cdot \int e^{2 x + 3} sin (4 x) d x))

Isolate

\int e^{2 x + 3} sin (4 x) d x

= - \frac{e ^{2 x + 3} cos ( 4 x )}{5} + \frac{e ^{2 x + 3} sin ( 4 x )}{10}

Add a constant to the solution

= - \frac{e ^{2 x + 3} cos ( 4 x )}{5} + \frac{e ^{2 x + 3} sin ( 4 x )}{10} + C

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What is the integral of e^{2x+3}*sin(4x) ?
The integral of e^{2x+3}*sin(4x) is -(e^{2x+3}cos(4x))/5+(e^{2x+3}sin(4x))/(10)+C