anonymous
  • anonymous
can someone help me solve an integral?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
|dw:1326769120780:dw|
Akshay_Budhkar
  • Akshay_Budhkar
the thing that immediately comes to my mind is use the \[\int\limits_{}^{} uvdx\] formula twice
anonymous
  • anonymous
ok in the midst of doing that

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Akshay_Budhkar
  • Akshay_Budhkar
but that will be a lengthy business, must have a shorter route
anonymous
  • anonymous
so what do u suggest?
anonymous
  • anonymous
cos(x)= Re[e^(i x)] x= e^(ln x) \[\int e^{ln(x)} e^x e^{i x} dx\]
anonymous
  • anonymous
\[\int e^{ln(x)+x+i x} dx\]
Akshay_Budhkar
  • Akshay_Budhkar
Complex numbers!! y didnt i think of that!
anonymous
  • anonymous
huh what is e^ix?
anonymous
  • anonymous
cos(x)= Re[e^(i x)] what is this step?
anonymous
  • anonymous
Using integration by parts three times to solve it isn't that bad either. It would take a while to type out all of the equations, but if you use 1. u=xe^x, dv=cos(x), then 2. u=xe^x, dv=sin(x), then 3. u=e^x, dv=cos(x) over your three stages, you will get the answer of \[\int xe^x\cos x dx = \frac{1}{2}xe^x(\sin x + \cos x) - \frac{1}{2}e^x\sin x + C\]
anonymous
  • anonymous
Eh. I would hesitate using complex numbers if you don't know how, you can get yourself into trouble.... using the natural extension of integration by parts to three functions is simple and straightforward, and makes sense already.
Akshay_Budhkar
  • Akshay_Budhkar
@imperialist that will be a bit of lengthy business though, if she knows complex numbers it will be faster
anonymous
  • anonymous
@Jemurray3, I agree. It would take far longer to explain why you can use complex numbers here (and even longer to understand it!) than 3 applications of integration by parts.
anonymous
  • anonymous
Thanks guys :D
anonymous
  • anonymous
I am just gonna wait to see what myin has to say for herself
myininaya
  • myininaya
First lets evaluate: \[\int\limits_{}^{}e^x \cos(x) dx\] \[=e^x \cos(x)-\int\limits_{}^{}e^x(-\sin(x)) dx=e^x \cos(x)+\int\limits_{}^{}e^x \sin(x) dx\] \[=e^x \cos(x)+(e^x \sin(x)-\int\limits_{}^{}e^x \cos(x) dx)\] So we have \[\int\limits_{}^{}e^x \cos(x) dx=e^x \cos(x)+(e^x \sin(x)-\int\limits_{}^{}e^x \cos(x) dx)\] So we shall solve this for: \[\int\limits_{}^{}e^x \cos(x) dx\] So we have \[\int\limits_{}^{}e^x \cos(x) dx=\frac{1}{2}e^x \cos(x)+\frac{1}{2}e^{x}\sin(x)+c_1\] ok now let's look back at \[\int\limits_{}^{} x e^x \cos(x) dx\] Apply integration by parts again! :) \[=(\frac{1}{2}e^x \cos(x)+\frac{1}{2}e^x \sin(x))x-\frac{1}{2}\int\limits_{}^{}(e^x \cos(x)+e^x \sin(x)) dx\] So we still need to find \[\int\limits_{}^{}e^x \sin(x) dx\] So we will need integration by parts again \[=e^x \sin(x)-\int\limits_{}^{}e^x \cos(x) dx\] \[=e^x \sin(x)-(e^x \cos(x)-\int\limits_{}^{}e^x (-\sin(x)) dx)\] So we need to solve this for: \[\int\limits_{}^{}e^x \sin(x) dx\] \[\int\limits_{}^{}e^x \sin(x) dx=\frac{1}{2} e^x \sin(x)-\frac{1}{2}e^x \cos(x)+ c_2\] So we need to put all of this together! lol
myininaya
  • myininaya
\[\int\limits_{}^{}x e^x \cos(x) dx=\frac{1}{2}x e^x \cos(x)+\frac{1}{2}x e^x \sin(x)-\frac{1}{2}(\frac{1}{2}e^x \cos(x)+\frac{1}{2}e^x \sin(x)+\] \[\frac{1}{2}e^x \sin(x)-\frac{1}{2}e^x \cos(x))+C\] after adding up all the constants I get C lol because the sum of some constants is still constant I think I got every part in there
anonymous
  • anonymous
LOL WOW
anonymous
  • anonymous
Thanks myin but u did it differently then what the others suggested
anonymous
  • anonymous
I wish I could give you a dozen medals myininaya for typing all of that up! :)
Akshay_Budhkar
  • Akshay_Budhkar
i tell her that for every answer! she is more than cool! :D
Akshay_Budhkar
  • Akshay_Budhkar
just tell ur kids not to trouble me @ myin :P :P
anonymous
  • anonymous
Thanks myin imperialist and jemurray:D I will have to review th steps now :D
myininaya
  • myininaya
Great question.
myininaya
  • myininaya
You guys are really sweet! :)
Akshay_Budhkar
  • Akshay_Budhkar
ur kids are sweet too :D
Akshay_Budhkar
  • Akshay_Budhkar
so are you :D
anonymous
  • anonymous
Oh myin u deserve the medal not me LOL
myininaya
  • myininaya
I don't have kids lol
myininaya
  • myininaya
you have kids not me lol
anonymous
  • anonymous
Akshay is like1o years younger than you LOL
Akshay_Budhkar
  • Akshay_Budhkar
yea i told her that she doesnt get me! her kidds come to my bakery and dont pay me
anonymous
  • anonymous
LOL and eat ur trash
Akshay_Budhkar
  • Akshay_Budhkar
LOL junk to be precise
myininaya
  • myininaya
lol
Akshay_Budhkar
  • Akshay_Budhkar
myin is blushing when i told her she has kids :P
anonymous
  • anonymous
lol she is dating satellite LOL hehe
Akshay_Budhkar
  • Akshay_Budhkar
!
Akshay_Budhkar
  • Akshay_Budhkar
satellite is 40 years old!
myininaya
  • myininaya
You guys are crazy. I never met satellite in person.
anonymous
  • anonymous
lol i was teasing
myininaya
  • myininaya
i know
Akshay_Budhkar
  • Akshay_Budhkar
rld study dont tease :P :P
anonymous
  • anonymous
yup got loads of work to do. Thanks myin U r awesome
Akshay_Budhkar
  • Akshay_Budhkar
what about me?
anonymous
  • anonymous
LOL dont even have to say that
Akshay_Budhkar
  • Akshay_Budhkar
yea yea~ :P i m so cool :P
anonymous
  • anonymous
if u live in my city = cool
Akshay_Budhkar
  • Akshay_Budhkar
later=D
anonymous
  • anonymous
ya bye guys
anonymous
  • anonymous
myin helped me solve it but i know u like integrals. The second I posted it u disappeared LOL Too bad
anonymous
  • anonymous
My bad, I'll help on the next one.
anonymous
  • anonymous
ya but now i am doing stupis things. I gotta refer to the tables of integrands and solve accordingly. Soon I will be solving using trig subsitution so i will be back :D

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