Question

how does the integral of [1/(x-1)(x+3)]dx equal [2/3(ln((x-1)/(4-x))+ ln2]?

Answer 1

Hi there :) welcome to OpenStudy!
\[\Large\rm \int\limits \frac{1}{(x-1)(x+3)}dx\]This requires Partial Fraction Decomposition.

Answer 2

\[\Large\rm \frac{1}{(x-1)(x+3)}=\frac{A}{(x-1)}+\frac{B}{(x+3)}\]Multiplying through by the denominator on the left gives us,\[\Large\rm 1=A(x+3)+B(x-1)\]

Answer 3

Any confusion up to that point? :o

Answer 4

no, I understand so far,:)

Answer 5

What else do I do?

Answer 6

Do you know how to solve the partial fraction?

Answer 7

yes I know how to solve the partial fraction and when I solve it I get something like 1-4(ln (x-1)/(x+3))

Answer 8

well, for the partial fractions I get (1/(4(x-1))-(1/(4(x+3)))

Answer 9

Were your values for A and B in the partial fractions 1/4 and -1/4 respectively?

Answer 10

yes

Answer 11

Ok that's correct.
Now if we integrate that we get
1/4ln(x-1)-1/4ln(x+3)
So your integral is correct i believe...

Answer 12

Wolfram also gives that.
https://www.wolframalpha.com/input/?i=integrate+1%2F%28%28x-1%29%28x%2B3%29%29

Answer 13

yes, but the answer is not available. the correct answer is (2/3)(ln((x-1)/(4-x))+ln2)

Answer 14

the problem says something like:
we find A,B so that 2/(x-1)(4-x)=(A/(x-1))+(B/(4-x))

Answer 15

please somebody help me find the answer?:(

Answer 16

that really confuses me, I don't know how they got that

Answer 17

That looks completely different from this problem 0_o

Answer 18

You sure you're matching up the correct answer? D:

Answer 19

yes, they were the corrections posted after I answered the problem and I didnt understand how they came up with it, I can try to upload a picture?

Answer 20

That would probably be a good idea :3

Answer 21

okay

Answer 22

sorry about the bad quality, I was not able to screenshot it so I took a picture with my phone

Answer 23

So here is the answer key 0_o
Hmmmm... no picture of the problem? :c

Answer 24

oh okay yes Ill do that

Answer 25

How come the problem is different to the working steps shown in the photo?

Answer 26

\( \displaystyle \int \dfrac{1}{(x-1)\color{#00aa00}{(x+3)}} \ dx \ \color{red}\ne \ \dfrac{2}{3} \ln \left( \dfrac{x-1}{\color{#aa0000}{4-x}} \right) + c \)

@ilymartin , I think the issue here is a typographical error. The 'correct' answer provided does not match the integral you began with; it appears to stem from a different integrand. Since you say this was provided after answering the question and should match, I suspect the issue was in the service and not in your Math. :)