inkyvoyd
Integrate
\(\Huge \frac{x}{ax^2+bx+c}\)
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waterineyes
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Linear by Quadratic...
inkyvoyd
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Eh?
inkyvoyd
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But, how exactly am I supposed to do this one?
waterineyes
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If you put ax2 + bx + c = y then it helps or not??
inkyvoyd
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uhh... what about the top - we just substitute using the quadratic formula in that case?
inkyvoyd
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Sorry guys, I'm not even sure where to start with this integral. I can't find it in my book either >.<
eyust707
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bring everything on the bottom to the top... then integration by parts?
inkyvoyd
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uhh - I'm not sure that will work >.<
eyust707
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why not...
inkyvoyd
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All the related integrals had something to do with trig sub and inverse trig functions, but i can't find this case.
eyust707
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ohh
inkyvoyd
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I know what the answer is (from an integral table), and it's logarithmic+trigonometric. I have no idea where to start though
eyust707
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ahhh that got me thinkin... let me take another look
eyust707
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well log came from 1/x
inkyvoyd
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Yes.
eyust707
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that looks like a big u sub
inkyvoyd
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Uh, got any ideas where to start htough?
eyust707
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one sec
inkyvoyd
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kay
eyust707
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just by the looks w/o really thinking about it, it looks like they let the bottom = u
eyust707
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then factor out a 2a
eyust707
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1/2a *
eyust707
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wait wtf
eyust707
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lol nvm i thought the second half was the next step
eyust707
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thats a tricky one i need a piece of paper for that one lol
inkyvoyd
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okay lol. Thanks for helping :P
eyust707
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that way is crazzyyyy tho lol
inkyvoyd
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lol - but the answer is different from the integral table :S the result is too complicated lol. I'm using some constants for the integral that will make it even more nasty x.x
eyust707
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its the same just written different
experimentX
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x = 1/2(2x + b/a) - 1/2 (b/a)
make two fractions ... and use the usual approach
eyust707
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^^^ thats what w.a did too
inkyvoyd
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Okay. any way to reduce it to the form seen in the integral table?
eyust707
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probably not
inkyvoyd
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once i substitute my values in, it is still an intermediate result - I still have to do two more substitutions, and I'm not sure I can do the algebra without going insane.
experimentX
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one is going to be log
another ... complete squares (a+sqrt(b/4a))^2 - (b/4a-c)
experimentX
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this is in standard form ... you should be able to evaluate this.
inkyvoyd
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Wait, I can do it by completing the square as well?
experimentX
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\[ \frac1{2a}\frac{x + b/a}{x + b/a x+c/a} - \frac1{2a}\frac{ b/a}{(x + \sqrt{b/4a})^2 - (b/4a-c/a)}\]
inkyvoyd
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oh wow. this integral - sqrt(tan x) - is a freaking pain...
experimentX
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quite a bit ... assuming c/a > b/4a
eyust707
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yea just about all realistic functions are either extremely tedious or impossible to solve by hand. When it comes to practical application, we usually just have computers solve or approximate the solution.
inkyvoyd
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Ugh, I give up. I'm tired of this integral lol. I'll look into it more tomorrow. Thanks though, @experimentX and @eyust707
eyust707
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thats when the mathematicians come in handy =P
eyust707
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yw inky.. you taking calc 2 over the summer??
inkyvoyd
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I'm trying to finish calc 3 and linear algebra and diff equations over the summer >.<
eyust707
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crazyyy
eyust707
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that is a lot of material to learn!!
inkyvoyd
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Yea, hopefully I can finish it.
eyust707
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I got faith!! where about are you in diffs?
inkyvoyd
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Haven't even started LOL
eyust707
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haha I see I see
inkyvoyd
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well, I skimmed the variables seperatble and the first order linear ODE I think
experimentX
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It's just a substitution ... reduces to the form
\[ \int \frac1 {x^2 + a^2}dx\]
or,
\[ \int \frac 1{x^2 - a^2}dx\]
assuming first, wolf calculates 2/a arctan(x/a)
eyust707
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yea seps will seem very natural
eyust707
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linear first order with the integrating factor is tricky to understand at first... but easy to do
eyust707
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youll prob see frustrating word problems but dont let them scare you.. they always work out to be quite simple
inkyvoyd
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alright - I have to learn how to integrate these nasty partial fractions first though haha :P
eyust707
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haha just remember, If your integrating with respect to y , all the other variables are constant!
eyust707
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keep repeating that in your head as you run through your derivative..
eyust707
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pretend that all the other letters are just numbers and youll be golden!
dpaInc
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didn't you see what i last posted earlier inky?
inkyvoyd
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uhh what did you post?