Question

Anybody know how to do this integral?

Answer 1

\[\int\limits \frac{ x ^{2} +1}{ (x-3)(x-2)^{2}}dx\]

Answer 2

I'd immediately try Partial Fractions. The integrand can be broken up into three fractions that look like this:

A B C
----- + ------ + --------
x - 3 x-2 (x-2)^2

because that (x-2)^2 produces a "repeated root."

I have little idea of how much experience you have with this topic, so will leave it to you to ask for clarification of anything that is not obvious to you here.

Answer 3

im having trouble finding the A, B and C

Answer 4

Have you found these coefficients before, for simpler cases, such as

A B
---- + ---- ?
x-1 x+2

Answer 5

i have that part but my problem is how to find A, B and C

Answer 6

Would you summarize, quickly, what you'd do to find A and B if

5x A B
---------- = ---- + ---- ?
(x-1)(x+2) x-1 x+2

Answer 7

this is a simpler problem than the one you've posted; I need to find out how much you already know about partial fraction expansions before going into detail about how to solve your posted question.

Answer 8

i know you get an x and like that you can find the A, B, C for my problem my first x was 2 since it would cancel A and B because is 0 and C=-5 but idk if im doing it right

Answer 9

I was hoping you'd discuss my sample problem first, but see that you are trying to redirect this conversation back to your posted problem.

Answer 10

\[\frac{ x ^{2+1} }{ (x-3)(x-2)^{2} }=\frac{ A }{ x-3 }+\frac{ B }{ x-2 }+\frac{ C }{(x-2) ^{2}}\]

Answer 11

for your problem is 5x=A(x+2)+B(x-1)

Answer 12

Yes. OK. So now I see you understand that part.
You'll see I've posted an equation above. In the first (given) fraction, the numerator should be x^2+1. What's the next step?

Answer 13

Hint: come up with an equation that looks like\[x ^{2}+1=A( ?)^{2}+B( ? )( ? ) + C( ?)\]

Answer 14

x^2+1=A(x-2)^2+B(x-2)(x-3)+C(x-3)

Answer 15

that's great! What would you suppose the next step would be? Personally, I'd let x=2 to find C.

Answer 16

is C=-5

Answer 17

And that confirms your earlier result for C: c=-5. Nice work. Do you need any further help from me, or can you now finish this problem on your own?

Answer 18

what x can i use to find B and A, i need help on that

Answer 19

You tried before to find A, B and C, and came up with values back then. What x-values did you use, and why?

Answer 20

Try this:

x^2+1=A(x-2)^2+B(x-2)(x-3)+C(x-3) (from you)

Replace C by -5 (which you found yourself)

Let x=2. Find A.

Let x=0 (arbitrary choice). Find B.

Enjoyed working with you. However, I really need to get to bed.
If you have further questions about the solution of this problem, reply here or private message me; I'll respond in the morning if need be. Good night!

Answer 21

is B=1/6 and to find A with x=2 it gives me 0

Answer 22

As before: I'll be back in the morning; cannot respond now.