How do you integrate (x)/(x-1)

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In summary, the conversation discusses different methods for integrating S x/(x-1) dx, including using Integration by Parts and substitution. The final solution involves using the substitution u=x-1, which simplifies the integral to \int \frac{u+1}{u}du and can be solved easily from there.
  • #1
amanda_ou812
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Homework Statement


How do you integrate S x/(x-1) dx


Homework Equations



IBP

The Attempt at a Solution


I tried using Integration by Parts with u= x/(x-1) and dv = dx and that did not work out. Then I tried using u= 1/(x-1) and dv = xdx but that did not work either.

A google search said "Just use the substitution u = x+1, then replace dx with du and you get u+1/u = 1 + 1/u which you can integrate to give u + ln u, thus = x-1 + ln(x-1)". But I am pretty sure that you can't just add +1 to the numerator like that.

I also know that it can be done using tables but is there a way not to use tables? Did I do my integration wrong?

Thanks
 
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  • #2
You don't add +1 to the numerator. We do the substitution u=x-1. What happens if you do that substitution??
 
  • #3
Hint Hint:
[itex]\frac{1}{x-1}[/itex]+[itex]\frac{x-1}{x-1}[/itex]=[itex]\frac{x}{x-1}[/itex]
 
  • #4
You can even do it without substitution by changing the equation to (x)/(x-1) = (x-1+1)/(x-1) = 1+1/(x-1), then integrating to get x+ln(x-1)+C.

Wow! All 3 posts within a minute!
 
  • #5
Perhaps this will make it clearer,

Suppose [itex]u=x-1[/itex]. Then clearly [itex]du=dx[/itex]. But notice also that [itex]u+1=x[/itex] by the first equation. Therefore, the integral becomes,

[itex]\int \frac{u+1}{u}du[/itex]

Now, it is trivial. :)
 
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  • #6
daveb said:
Wow! All 3 posts within a minute!

We're all just so stoked about integration, we can't resist. If only all students shared our passion for integration... ;)
 

1. How do you solve the integral of (x)/(x-1)?

The integral of (x)/(x-1) can be solved using the method of partial fractions. This involves breaking down the fraction into simpler fractions and then integrating each part separately.

2. Can you provide an example of how to integrate (x)/(x-1)?

Yes, for example, the integral of (x)/(x-1) can be solved as follows:
(x)/(x-1) = (1+1/(x-1))
= (1+1/(x-1))dx
= (1+1/u)du (where u = x-1)
= u+ln(u) + C (where C is the constant of integration)
= (x-1) + ln(x-1) + C
= x + ln(x-1) + C

3. What are the steps involved in integrating (x)/(x-1)?

The steps involved in integrating (x)/(x-1) are:
1. Rewrite the fraction as a sum of simpler fractions using the method of partial fractions.
2. Integrate each part separately.
3. Substitute back the original variable.
4. Add the constant of integration.

4. Is there a shortcut for integrating (x)/(x-1)?

No, there is no shortcut for integrating (x)/(x-1). It requires the method of partial fractions to be solved.

5. How do I know if I have integrated (x)/(x-1) correctly?

You can verify your answer by differentiating it. If you get back the original function (x)/(x-1), then your integration is correct. You can also use online integration calculators to check your answer.

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