Integral e^x cos x two ways

In this awesome integral video, I calculate the integral of e^x cos x in two ways, one using integration by parts, and the other using complex numbers. Which method do you like more? Watch this video and find out! Enjoy :)

[qubix]

I really hate integrals e^(ax)cos(bx) and e^(ax)sin(bx) (in russian we call these "cyclic integrals") because I know there is a general formula for these two but it's pretty complicated, and every time I face them I can never remember it, so I have to go through 2 boring integrations by parts yet again. This "OMG" method can really speed me up, thanks

[SlingerDomb]

Euler's identity for our integral! How cool is that!
I've seen the latter method several times already but it is still so impessive nontheless.

[emman]

He made the integral easier through complexification! The first time I read such a sentence, I thought the textbook I was reading made an error but no, complexification apparently makes things easier lol. 😂

[calvinkywong]

Me watching this video
**Oh my god he speaks in English!!!*

[Green_Eclipse]

Nice! I watched for the second method and it was definitely worth it.

[LS-Moto]

You're love for math is so contagious :)

[robsbackyardastrophotograp]

That second method was amazing! Two for the price of one!

[JJCUBER]

I can’t wait for my AP Calc AB class to get to integrals in 1 week! I’ll already know like 90% of the stuff you need to know!

[shambosaha]

I like how he just assumes that I am impressed while I am actually like "Whoaaaa..."

[musik]

Roses are red,
π should sound like pee,
you're not the first one
to forget the +C.

[immutabledestiny]

Is it possible to integrate this by taking the integral of the product of the Taylor-maclaurin series of e^x and cos(x) respectively

[danielmilyutin]

Hi Dr Peyam, are there some non-standard unusual ways to do this integral?

[mariopina]

Oh no, the evil attack of constants :0

[holyshit]

Yes but on my calculus for IT class we had only single variable calculus with real numbers (we had a little bit of complex numbers on algebra)
so only by parts method would be accepted

[bavrined]

Hello, dear Dr. Peyam! Could you please record a video on Riemann's zeta-function, something like an introduction to the theme, main aspects, an abstract? Thanks in advance.

[alwysrite]

love when you use the word "junk" -- Doctor of Math calling it "junk" : )

[Peter_]

What would the derivative for y(x) = sin(x) be if you took the infinitieth derivative?

[juauke]

You forgot the +C :o for the integration by parts method ! Still a great video (my favorite way is definitely the complexification of the integral :D) ^^ !

[HawluchaMCPE]

Thank you for inspiring me and showing me new ways, Dr. Peyam :) however I am now turning off notifications from your channel because I think my life can be somewhat better by reducing the time I spend watching youtube and instead focusing on educational material. I hope it goes well. Sorry for that, but I think that I just have to do that now. Your content is great and has quite inspired me in situations I wouldn't like; for example, in your f(f(f(x)))=f(x) video I would just call myself braindead and leave the question. You got to the quirkiest of topics. Thank you for all this. ❤

Also I have a question: prove or disprove that all circles have the same number of points.

[EpicMathTime]

Ah, ok, so

int e^x cos(x) dx =
int e^x Re(e^ix) dx =
Re(int e^((1+i)x) dx) =
Re((e^(1+i)x)/(1+i)) =
0.5 Re(e^x * e^ix - ie^x * e^ix) =
0.5 e^x Re(e^ix - ie^ix) =
0.5 e^x (cos(x) + sin(x))
👏