anonymous
  • anonymous
Integral of x^2 dx/x^2 + 9 Use partial fraction decomposition, and long division if needed.
Calculus1
  • Stacey Warren - Expert brainly.com
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SOLVED
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katieb
  • katieb
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
SolomonZelman
  • SolomonZelman
I will post an example
SolomonZelman
  • SolomonZelman
For example: \(\large\color{black}{\displaystyle\int\limits_{~}^{~}\frac{x^2}{x^2+4}~dx}\) You will perform this trig-sub: \(\large\color{black}{\displaystyle x=\sqrt{4}\tan\theta=2\tan\theta}\) \(\large\color{black}{\displaystyle dx/d\theta =2\sec^2\theta\quad\Longrightarrow\quad dx=2\sec^2\theta{~}d\theta }\) \(\large\color{black}{\displaystyle\int\limits_{~}^{~}\frac{\left(2\tan\theta\right)^2}{\left(2\tan\theta\right)^2+4}~\left(2\sec^2\theta{~}d\theta\right)}\) \(\large\color{black}{\displaystyle\int\limits_{~}^{~}\frac{4\tan^2\theta}{4\tan^2\theta+4}~\left(2\sec^2\theta{~}d\theta\right)}\) \(\large\color{black}{\displaystyle\int\limits_{~}^{~}\frac{8\tan^2\theta\sec^2\theta}{4\tan^2\theta+4}~{~}d\theta}\) \(\large\color{black}{\displaystyle\int\limits_{~}^{~}\frac{8\tan^2\theta\sec^2\theta}{4(\tan^2\theta+1)}~{~}d\theta}\) \(\large\color{black}{\displaystyle\int\limits_{~}^{~}\frac{8\tan^2\theta\sec^2\theta}{4(\sec^2\theta)}~{~}d\theta}\) \(\large\color{black}{\displaystyle\int\limits_{~}^{~}2\tan^2\theta{~}d\theta}\) \(\large\color{black}{\displaystyle\int\limits_{~}^{~}2\sec^2\theta+2{~}d\theta}\) \(\large\color{black}{\displaystyle 2\tan\theta+2\theta+C}\) Now we will rearrange \(\theta\) in terms of x: \(\large\color{black}{\displaystyle x=2\tan\theta}\) \(\large\color{black}{\displaystyle \frac{x}{2}=\tan\theta}\) \(\large\color{black}{\displaystyle \theta=\arctan\left(\frac{x}{2}\right)}\) Now do the substitution: \(\large\color{black}{\displaystyle 2\tan\left(\arctan\left(\frac{x}{2}\right)\right)+2\arctan\left(\frac{x}{2}\right)+C}\) \(\large\color{black}{\displaystyle 2\cdot\frac{x}{2}+2\arctan\left(\frac{x}{2}\right)+C}\) \(\large\color{black}{\displaystyle x+2\arctan\left(\frac{x}{2}\right)+C}\)
SolomonZelman
  • SolomonZelman
I just used a 4 instead of a 9, but I hope that this helps.

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anonymous
  • anonymous
Trig substitution, yeah... I'm just not sure if I'm supposed to solve the problem with just the long division/partial fractions. I guess not, if you're not seeing a way to do it that way.
SolomonZelman
  • SolomonZelman
I don't think you can do partial fractions on this because the bottom can not be factored. Trig sub will be needed
SolomonZelman
  • SolomonZelman
\(\rm x^2+4\) is not factorable (well, not into reals)
anonymous
  • anonymous
Right. Thanks!
SolomonZelman
  • SolomonZelman
You Welcome !
ℐℵK℣ⱱøƴȡ◆
  • ℐℵK℣ⱱøƴȡ◆
this is wrong
SolomonZelman
  • SolomonZelman
Are you sure that this is wrong? I really doubt that there is any error in my example, although anyone can make a mistake. If you think that I really made a mistake, then please pint out where and how.
ℐℵK℣ⱱøƴȡ◆
  • ℐℵK℣ⱱøƴȡ◆
nooooooooooooooo
ℐℵK℣ⱱøƴȡ◆
  • ℐℵK℣ⱱøƴȡ◆
you're fine
SolomonZelman
  • SolomonZelman
Then what is wrong? To do what I did is a violation of the site's policy?
ℐℵK℣ⱱøƴȡ◆
  • ℐℵK℣ⱱøƴȡ◆
wat o.O
ℐℵK℣ⱱøƴȡ◆
  • ℐℵK℣ⱱøƴȡ◆
my love for you is wrong ;-; k bye
SolomonZelman
  • SolomonZelman
\(\Large\color{black}{\displaystyle\lim_{a \rightarrow ~\infty}\left( x{\rm D} \right)^a}\)
ℐℵK℣ⱱøƴȡ◆
  • ℐℵK℣ⱱøƴȡ◆
awh :') ofc our love is beyond infinity <3

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