anonymous
  • anonymous
Evaluate the integral using integration by parts w/ the indicated choices of u and dv how do i do this? equation inside! thanks!!
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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schrodinger
  • schrodinger
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anonymous
  • anonymous
|dw:1442193899384:dw|
thomas5267
  • thomas5267
So the equation is: \[ \int\theta\cos(\theta)\,d\theta\\ u=\theta\\ dv=\cos(\theta)\,d\theta\\ \int u\,dv=uv-\int v\,du \]
anonymous
  • anonymous
yes:) i'm a bit confused, so i find du and v, right? would du = ø and v = -cosødø ?

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zepdrix
  • zepdrix
Hmm if \(\large\rm u=\theta\) then \(\large\rm du\ne\theta\) You're taking a derivative with respect to theta :o \(\large\rm u'=1\) yes?
anonymous
  • anonymous
yes :) ohh so du = 1 ?
zepdrix
  • zepdrix
\[\large\rm \color{orangered}{u'}=1\]\[\large\rm \color{orangered}{\frac{du}{d\theta}}=1\qquad\to\qquad du=1\cdot d\theta\]
anonymous
  • anonymous
ohh okay :) so which means du = dø ? and so would v be sinø ? :/
zepdrix
  • zepdrix
To get from \(\large\rm dv\) to \(\large\rm v\) you have to integrate. So going backwards from cosine gives us sine, ya that sounds right! :)\[\large\rm dv=\cos\theta~d\theta\qquad\to\qquad v=\sin\theta\]
anonymous
  • anonymous
ooh yay!! :) and so now i do this?|dw:1442194685139:dw|
zepdrix
  • zepdrix
I wish you wouldn't use the "empty set" for your theta XD lol but ya looks good so far!
anonymous
  • anonymous
ohh haha oops :P it's just more convenient using that symbol instead of drawing it :P and so would it get sin2ø - (-cosø)(ø) ?
anonymous
  • anonymous
getting sin2ø + cos2ø + c ?
zepdrix
  • zepdrix
Hmm
zepdrix
  • zepdrix
In general: \(\large\rm a\cos(x)\ne\cos(ax)\) You can't just bring stuff inside of the trig function willy nilly like that.
anonymous
  • anonymous
ohh okay :( how do i do this part then? |dw:1442195029862:dw|
zepdrix
  • zepdrix
We leave this alone \(\large\rm \theta\sin\theta\) that will be part of our final answer
zepdrix
  • zepdrix
\[\large\rm -\int\limits\sin\theta~d\theta\]Hmm looks like we made some kind of boo boo here on this integral.
zepdrix
  • zepdrix
If you ignore the negative out front, what is the integral of sine?
anonymous
  • anonymous
cos ?
zepdrix
  • zepdrix
Hmm, no we're going backwards. Going forwards, derivative of sine is cosine.
anonymous
  • anonymous
ohh - cos ?
zepdrix
  • zepdrix
\[\large\rm -\color{orangered}{\int\limits\limits\sin\theta~d\theta}=\quad -\color{orangered}{(-\cos \theta+c)}\]Mmm ya that sounds better!
zepdrix
  • zepdrix
So you have:\[\large\rm \int\limits \theta \cos \theta~d \theta=\theta \sin \theta-(-\cos \theta+c)\]
anonymous
  • anonymous
ohh okay and so i simply it to øsinø + cos ø - c ?
zepdrix
  • zepdrix
Or even: \(\large\rm \theta\sin\theta+\cos\theta+C\) absorb the negative into the c, since it can represent any negative or positive value.
anonymous
  • anonymous
ahh okay! thank you!!:)
zepdrix
  • zepdrix
yay team \c:/

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