Complex Integral

[dcl]

Here is the problem:
{\mathop{\rm Im}\nolimits} \int {e^{x(2 + 3i)} } dx

One sec, I'm having another go at it.

= {\mathop{\rm Im}\nolimits} \int {e^2 } e^{3ix} dx

= {\mathop{\rm Im}\nolimits} \int {e^2 } [\cos (3x) + i\sin (3x)]dx


\begin{array}{l}
= \frac{{ - e^2 \cos (3t)}}{3} \\
\end{array}


How'd I go?

[arildno]

Not very well, I'm afraid..
Let us first consider the problem to calculate the antiderivative of the complex exponential,
\int{e}^{(2+3i)x)}dx

This is simply:
\int{e}^{(2+3i)x)}dx=\frac{1}{2+3i}{e}^{(2+3i)x)}+ C

where C is an arbitrary complex constant (I'll set it in the following to 0, for simplicity)

We are to find the imaginary part:
Im(\frac{1}{2+3i}{e}^{(2+3i)x)})=Im(\frac{2-3i}{13}{e}^{(2+3i)x)})
or:
Im(\frac{1}{2+3i}{e}^{(2+3i)x)})=\frac{e^{2x}}{13} (2\sin(3x)-3\cos(3x))

[Zurtex]

e^{x(2+3i)} = e^{2x}e^{3xi} \neq e^{2}e^{3xi}

It's easier than that. Remember that:

\int e^{ax} dx = \frac{1}{a}e^{ax} + C

Once you integrate you then need to separate real from imaginary.

[dcl]

Stuffed up in the first step :(
Thanks guys. :)
prolly should goto bed now. :(