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

The integral of e^{2x}sin(x/2) is 8/17 e^{2x}sin(x/2)-2/17 e^{2x}cos(x/2)+C

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

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

\frac{8}{17} e^{2 x} sin (\frac{x}{2}) - \frac{2}{17} e^{2 x} cos (\frac{x}{2}) + C

Solution steps

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

Apply Integration By Parts

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

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

\int \frac{1}{4} e^{2 x} cos (\frac{x}{2}) d x = \frac{1}{34} (e^{2 x} sin (\frac{x}{2}) + 4 e^{2 x} cos (\frac{x}{2}))

= \frac{1}{2} e^{2 x} sin (\frac{x}{2}) - \frac{1}{34} (e^{2 x} sin (\frac{x}{2}) + 4 e^{2 x} cos (\frac{x}{2}))

Simplify

\frac{1}{2} e^{2 x} sin (\frac{x}{2}) - \frac{1}{34} (e^{2 x} sin (\frac{x}{2}) + 4 e^{2 x} cos (\frac{x}{2})) : \frac{8}{17} e^{2 x} sin (\frac{x}{2}) - \frac{2}{17} e^{2 x} cos (\frac{x}{2})

= \frac{8}{17} e^{2 x} sin (\frac{x}{2}) - \frac{2}{17} e^{2 x} cos (\frac{x}{2})

Add a constant to the solution

= \frac{8}{17} e^{2 x} sin (\frac{x}{2}) - \frac{2}{17} e^{2 x} cos (\frac{x}{2}) + C

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What is the integral of e^{2x}sin(x/2) ?
The integral of e^{2x}sin(x/2) is 8/17 e^{2x}sin(x/2)-2/17 e^{2x}cos(x/2)+C