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 
Senior Member Joined: Feb 2016 From: Australia Posts: 1,638 Thanks: 570 Math Focus: Yet to find out.  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. WolframAlpha: Computational Knowledge Engine 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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