Question

Integrate ( (x^2 -2x -1) / (x-1)^2(x^2 +1) ) dx

Answer 1

looks like a job for partial fractions, of the most annoying kind

Answer 2

but it has been cooked up in this case so all your coefficients are easy

Answer 3

I found some values for A B C and D but idk if they are right they give me a shady answer..

Answer 4

\[\frac{x^2-2x-1}{(x-1)^2(x^2+1)}\]
\[=\frac{A}{x-1}+\frac{B}{(x-1)^2}+\frac{Cx+d}{(x^2+1}\] is a start

Answer 5

did you get \[\frac{1}{x-1}+\frac{1}{(x-1)^2}+\frac{-x+1}{x^2+1}\]?

Answer 6

when you say A,B,C, D i am not sure where you put them, so it is not clear

Answer 7

if that is what you got, the integrals are easy enough

Answer 8

What are the values you got for the numerators?

Answer 9

the ones i wrote above

Answer 10

A,B,C,D I mean

Answer 11

like i said, it depends on how you label them
the answer is above

Answer 12

\[\frac{1}{x-1}+\frac{1}{(x-1)^2}+\frac{-x+1}{x^2+1}\]

Answer 13

It doesnt if you just tell which values you have numerically without mention of the alphabetical variables.

Answer 14

three ones and a minus one
which integral are you having trouble with?

Answer 15

I got different values but hold on Idk if I might get the right answer anyways...

Answer 16

Keep in mind that you can choose any letters you want for the numerators of these partial fractions, and the order in which you use them doesn't matter.

Note that the denominator of the integrand you were given factors out to (x-1)(x-1)(x^2+1).

There are 3 factors here. You'll end up with 3 partial fractions. Note that (x-1)^2 is a "repeated root." That's why (x-1) and (x-1)^2 show up in two of the three partial fractions.

Arbitrarily choosing letters for the numerators of these three partial fractions, we might get \[\frac{ P}{ x-1 }+\frac{ N }{ (x-1)^2 }+\frac{ Cx+F }{ x^2+1 }\]

Answer 17

Need any further help with this problem? Have you been able to find the values of the four constant coefficients, whether A, B, C and D or P, N, C and F?

Answer 18

no not yet

Answer 19

No, not yet (to finding the coeff. values), or no, not yet (to needing help)?

Answer 20

finding the coefficient values...

Answer 21

OK. Have you a preference for the method used to find the values of those coefficients?

\[\frac{ P}{ x-1 }+\frac{ N }{ (x-1)^2 }+\frac{ Cx+F }{ x^2+1 }=\frac{ x^2-2x-1 }{ (x-1)^2(x^2+1) }\]

Answer 22

One (of several available methods) is this: Choose any four constants, such as 0, -1, 2, 3. One at a time, substitute said values for every occurrence of "x" into the equation given above. Note that you can't use x=1 for this purpose because you'd then have division by zero.

Do this 3 more times, so you'll end up with four equations in four unknowns (P, N, C and F). Then this becomes a matrix problem where your job is to find the values of P, N, C and F.

Making progress? or need any further assistance?

Answer 23

If you were to let x=0, here's what your result would look like:
\[\frac{ P}{ 0-1 }+\frac{ N }{ (0-1)^2 }+\frac{ Cx+F }{ 0^2+1 }=\frac{ 0^2-2(0)-1 }{ (0-1)^2(0^2+1) }\]

Answer 24

Were you to let x=0, here's what you'd get:
\[\frac{ P}{ 0-1 }+\frac{ N }{ (0-1)^2 }+\frac{ C(0)+F }{ 0^2+1 }=\frac{ 0^2-2(0)-1 }{ (0-1)^2(0^2+1) }\]

Answer 25

You'd do this 3 more times, for x= -1, 2, 3.

Answer 26

Still interested in finishing this problem? It's been 16 minutes since I last heard from you.

Answer 27

I got A=-1/4 B=-1 C=1/4 and D=-1/4 so we have A/x-1 + B/(x-1)^2 + Cx+D / (x^2 +1)

Answer 28

You have come up with proposed numerical values for A, B, C and D. You need to substitute these numerical values for A, B, C and D in your expression on the far right:

\[(-1/4)\frac{ 1 }{ x-1 }-1\frac{ 1 }{ (x-1)^2 }+\frac{ -1 }{ 4 }\frac{ x }{ x^2+1 }-\frac{ 1 }{ 4 }\frac{ x }{ x^2+1 }\]

Answer 29

While this differs from the result satellite73 obtained and shared with you earlier, that doesn't mean you're not correct.

Any questions or comments at this point?

Answer 30

I've got an answer but how did you get your answer if you didn't get these values?

Answer 31

And the LCD is (x^2 +1 )(x-1)?

Answer 32

There are various methods in which to solve for the unknown coefficients. I don't know which method you used, nor do I know which method satellite73 used. My most recent post reflects YOUR work in that I substituted y our value for A into the expression with fractions, and so on.

So, what are you asking me?

Answer 33

I'm more interested in seeing / knowing that you know how to apply at least one of those methods for finding the coefficient values. Accuracy in applying it may come later.