- #1
HotMintea
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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]
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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