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 January 31st, 2018, 08:24 PM #1 Newbie   Joined: Jan 2018 From: U.S. Posts: 1 Thanks: 0 Evaluate an Integral ∫e^(-x)cos(x)dx Last edited by skipjack; February 8th, 2018 at 09:50 AM.
January 31st, 2018, 08:27 PM   #2
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Quote:
 Originally Posted by margaretk ∫e^(-x)cos(x)dx
If you aren't going to bother attempting the exercise for yourself, or explain any difficulties you are having, I suggest you pay for a Wolfram Alpha subscription and get worked solutions there.

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Last edited by skipjack; February 8th, 2018 at 09:51 AM.

 January 31st, 2018, 08:36 PM #3 Senior Member   Joined: Oct 2009 Posts: 492 Thanks: 164 Sure, sure, I know the usual way, but I would actually prefer to work out $$\int e^{-x} e^{ix}dx$$ instead. The solution follows from this, and you get one for free too.
 February 8th, 2018, 04:52 AM #4 Math Team   Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894 Specifically, what microm@ss is suggesting is that, since $\displaystyle \cos(x)= \frac{e^{ix}+ e^{-ix}}{2}$, $\displaystyle e^x \cos(x)= e^x\left(\frac{e^{ix}+ e^{-ix}}{2}\right)= \frac{1}{2}\left(e^{(1+ i)x}+ e^{1- i)x}\right)$. Can you integrate that? Another way, using integration by parts: Let $\displaystyle u= \cos(x)$, $\displaystyle dv= e^x dx$. Then $\displaystyle du= -\sin(x) dx$ and $\displaystyle v= e^x$. So, by "integration by parts" ($\displaystyle \int u dv= uv- \int vdu$) $\displaystyle \int e^x \cos(x)= e^x \cos(x)- \int (-\sin(x))e^x dx= e^x \cos(x)+ \int e^x \sin(x)dx$. To do that integral, use integration by parts again. Let $\displaystyle u= \sin(x)$ $\displaystyle dv= e^x dx$. Then $\displaystyle du= \cos(x)dx$ and $\displaystyle v= e^x$ so that $\displaystyle \int e^x \sin(x)= e^x \sin(x)- \int e^x \cos(x) dx$. Putting that into the previous integral, $\displaystyle \int e^x \cos(x) dx= e^x \cos(x)+ e^x \sin(x)- \int e^x \cos(x) dx$. Now, add $\displaystyle \int e^x \cos(x) dx$ to both sides to get $\displaystyle 2\int e^x \cos(x) dx= e^x \cos(x)+ e^x \sin(x)$ and then $\displaystyle \int e^x \cos(x) dx= \frac{1}{2}e^x(\cos(x)+ \sin(x))+ C$ Last edited by skipjack; February 8th, 2018 at 09:54 AM.
 February 10th, 2018, 08:47 PM #5 Newbie   Joined: Jan 2018 From: somewhere Posts: 14 Thanks: 2 Math Focus: Algebraic Number Theory / Differential Fork Theory For those not familiar with that fact, using integration by parts combined with a trig substitution may be useful here.

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