My Math Forum Two integration problems

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October 4th, 2012, 02:55 PM   #1
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Joined: Oct 2012

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Two integration problems

Hello.
I am struggling with the following two integrals.
Any help would be appreciated!

This requires trigonometric substitution.
[attachment=0:2nvy98bt]CodeCogsEqn5.gif[/attachment:2nvy98bt]

I believe this one has to be done with partial fraction decomposition.
[attachment=1:2nvy98bt]CodeCogsEqn(1).gif[/attachment:2nvy98bt]
Attached Images
 CodeCogsEqn(1).gif (1.0 KB, 85 views) CodeCogsEqn5.gif (826 Bytes, 84 views)

October 4th, 2012, 06:56 PM   #2
Math Team

Joined: Dec 2006
From: Lexington, MA

Posts: 3,267
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Re: Two integration problems

Hello, brh27!

Quote:
 $\int\frac{125x^2\,dx}{\sqrt{4\,-\,25x^2}}$

$\text{W\!e have: }\:125\int\frac{x^2\,dx}{\sqrt{4\,-\,25x^2}}$

$\text{Let }5x \:=\:2\sin\theta\;\;\;\Rightarrow\;\;\;x \:=\:\frac{2}{5}\sin\theta\;\;\;\Rightarrow\;\;\;d x \:=\:\frac{2}{5}\cos\theta\,d\theta$

$\text{Substitute: }\:125\int\frac{(\frac{4}{25}\sin^2\theta)(\frac{2 }{5}\cos\theta\,d\theta)}{2\cos\theta} \;=\;4\int\sin^2\theta\,d\theta$

[color=beige]. . . . . . . . [/color]$=\;2\int(1\,-\,\cos2\theta)d\theta \;=\;2(\theta\,-\,\frac{1}{2}\sin2\theta)\,+\,C \;=\;2(\theta\,-\,\sin\theta\cos\theta)\,+\,C$

$\text{Back-substitute: }\:\sin\theta \,=\,\frac{5x}{2},\;\;\;\cos\theta \,=\,\frac{\sqrt{4\,-\,25x^2}}{2}$

$\text{Answer: }\:2\,\left(\arcsin\frac{5x}{2} \,-\,\frac{5x\sqrt{4\,-\,25x^2}}{4}\right)\,+\,C$

Quote:
 $\int \frac{3x^3\,+\,2x^2\,+\,3x\,-\,9}{x^4\,+\,3x^2}\,dx$

$\text{W\!e have: }\:\frac{3x^3\,+\,2x^2\,+\,3x\,-\,9}{x^2(x^2\,+\,9)} \;=\;\frac{A}{x}\,+\,\frac{B}{x^2}\,+\,\frac{Cx\,+ \,D}{x^2\,+\,9}$

$\text{Then: }\:3x^3\,+\,2x^2\,+\,3x\,-\,9 \;=\;Ax(x^2+9)\,+\,B(x^2+9)\,+\,Cx^3\,+\,Dx^2$

$\text{I'll skip the gruesome details.}
\;\;\text{W\!e get: }\:A\,=\,\frac{1}{3},\;\;B\,=\,-1,\;\;C \,=\,\frac{8}{3},\;\;D\,=\,3$

$\text{Therefore: }\;\frac{\,\frac{1}{3}\,}{x}\,-\,\frac{1}{x^2}\,+\,\frac{\frac{8}{3}x}{x^2\,+\,9} \,+\,\frac{3}{x^2\,+\,9}$

$\text{Can you finish it?}$

 October 4th, 2012, 08:57 PM #3 Newbie   Joined: Oct 2012 Posts: 10 Thanks: 0 Re: Two integration problems Thank you! You're a life saver.

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