anonymous
  • anonymous
how to do this integral?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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chestercat
  • chestercat
I got my questions answered at brainly.com in under 10 minutes. Go to brainly.com now for free help!
anonymous
  • anonymous
\[\int\limits_{}\frac{ x^2 + 1 }{ (x-3)(x-2)^2}dx\]
misty1212
  • misty1212
HI!!
anonymous
  • anonymous
hello!!

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misty1212
  • misty1212
how are you at partial fractions?
anonymous
  • anonymous
i understand it a bit (split into parts, right? A/B?) but don't know quite how to apply it!
misty1212
  • misty1212
it is going to suck want to do it step by step, or cheat?
anonymous
  • anonymous
can you show me how to do it step by step?
misty1212
  • misty1212
\[\frac{A}{x-3}+\frac{B}{x-2}+\frac{C}{(x-2)^2}\]
anonymous
  • anonymous
okay! what next? :O
misty1212
  • misty1212
that means \[A(x-2)^2+B(x-3)(x-2)+C(x-3)=x^2+1\]
misty1212
  • misty1212
let me know if that is clear, then we can continue
anonymous
  • anonymous
yes:)
anonymous
  • anonymous
it is clear:) what happens next?
misty1212
  • misty1212
we can get C right away by putting \(x=2\) because the A and B will drop out
misty1212
  • misty1212
if you put \(x=2\) you get \[C(2-3)=2^2+1\] or \[-C=5\] making \(C=-5\)
misty1212
  • misty1212
got that?
anonymous
  • anonymous
yes:) sorry, OS connection was very poor!! what happens next?
IrishBoy123
  • IrishBoy123
get A by setting x = 3 same technique
anonymous
  • anonymous
okay, so then i get 3^2 + 1 = 10 ?
anonymous
  • anonymous
A=10?
IrishBoy123
  • IrishBoy123
\[A(3-2)^2+B(3-3)(3-2)+C(3-3)=3^2+1\] \[\checkmark\]
anonymous
  • anonymous
ok!! what do i do next?
IrishBoy123
  • IrishBoy123
\[10(x-2)^2+B(x-3)(x-2)-5(x-3)=x^2+1\] you need B so set x to anything really. but let's just go with x = 0.
anonymous
  • anonymous
okay! so B=1?
IrishBoy123
  • IrishBoy123
sure? 10(4) + B(-3)(-2) - 5(-3) = 1 6B = (1 -15 -40)????
anonymous
  • anonymous
ohh oops!! it equals -55!
anonymous
  • anonymous
-0.1111 = B?
anonymous
  • anonymous
@IrishBoy123 i have to leave for an appointment now, but will be back in a few hours!! is it okay if we continue then, or if i come back and see what you say?
IrishBoy123
  • IrishBoy123
\[\color{red}6B = -5 \color{red}4\]
IrishBoy123
  • IrishBoy123
sure
anonymous
  • anonymous
@IrishBoy123 I'm back!! can you please help me explain the rest?
anonymous
  • anonymous
so B = -9 right? what happens next?
IrishBoy123
  • IrishBoy123
let's agree where we got to first
IrishBoy123
  • IrishBoy123
\[\frac{10}{x-3}-\frac{9}{x-2}-\frac{5}{(x-2)^2}\] are we sure that's a good summary?
anonymous
  • anonymous
okay! so A=10, B=-9 and C=-5 ?
IrishBoy123
  • IrishBoy123
now you must do \[\int \frac{10}{x-3}dx- \int \frac{9}{x-2}dx-\int\frac{5}{(x-2)^2} dx\] 3 separate integrals
anonymous
  • anonymous
ooh okay, how do i start off?
IrishBoy123
  • IrishBoy123
first one \[\int \frac{10}{x-3}dx\]
anonymous
  • anonymous
okay, so we get 10lnx-3 ?
IrishBoy123
  • IrishBoy123
y second?
anonymous
  • anonymous
so it is not that?
IrishBoy123
  • IrishBoy123
\[\int \frac{9}{x-2}dx\]
anonymous
  • anonymous
or it is?
IrishBoy123
  • IrishBoy123
first part is right, do the second now
anonymous
  • anonymous
ohhh i see now and so now second we get 9lnx-2 ?
anonymous
  • anonymous
and for the third, we get -5/(x-2) ?
IrishBoy123
  • IrishBoy123
yes. just make a note that we need to put the correct signs between the integrals righ at the end as it is ∫dx - ∫dx - ∫dx last one \[\int\frac{5}{(x-2)^2} dx\]
IrishBoy123
  • IrishBoy123
yes put it all together now
IrishBoy123
  • IrishBoy123
don't bother writing it out if you are happy with it all
anonymous
  • anonymous
\[\int\limits_{}10lnx-3 dx - \int\limits_{}9lnx-2dx + \int\limits_{} \frac{ 5 }{ x-2 }dx\]
anonymous
  • anonymous
like that?
IrishBoy123
  • IrishBoy123
no, drop all the integration signs now that you have integrated so \[10 \ln(x-3) - 9\ln(x-2) + \frac{ 5 }{ x-2 }\]
IrishBoy123
  • IrishBoy123
OK?
anonymous
  • anonymous
ohh okay!! what now? :)
IrishBoy123
  • IrishBoy123
i think the fat lady is now singing.
anonymous
  • anonymous
hahaha ohh so we are done?
anonymous
  • anonymous
@IrishBoy123 ?
IrishBoy123
  • IrishBoy123
OMG!!!! we forgot something....
anonymous
  • anonymous
what?? :O
IrishBoy123
  • IrishBoy123
the integration constant!!
IrishBoy123
  • IrishBoy123
how embarassing after all that :-))
anonymous
  • anonymous
ohh + C ?
IrishBoy123
  • IrishBoy123
indeed!
anonymous
  • anonymous
ohh okie!! yay!! thank you so much!!
IrishBoy123
  • IrishBoy123
you and misty did all the heavy lifting, we just finished it off:) good night!
anonymous
  • anonymous
good night!:D

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