(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement

1.1. Is it possible to do [itex] \int\ sin{x}\, \ cos{x}\, \ e^x\, \ dx\ [/itex] by complexifying the integral? (Note: not by integration by parts.)

Complexifying the Integral (Arthur Mattuck, MIT) [9:23]

1.2. When is it appropriate to complexify an integral, beside the condition that the integrand can be expressed as [itex] Re (\ e^{\alpha x})\, \ [/itex]?

2. The attempt at a solution

2.1.

[tex] \begin{equation*}

\begin{split}

\int\ sin{x}\, \ cos{x}\, \ e^{x}\, \ dx\ =\\

\int\ Re(\ e^{i(\frac{\pi}{2}\ -\ x)}\ )\, \ Re(\ e^{ix})\, \ e^{x}\ dx\ =\\

Re\int\ e^{i(\frac{\pi}{2}\ -\ x)}\, \ e^{ix}\, \ e^{x}\ dx\ =\\

Re\int\ i\ e^x\, \ dx\ =\\

- Im( e^x )\ + \ C\, \, (?)

\end{split}

\end{equation*} [/tex]

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# Complexifying an integral.

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