Integrating a Fraction with a Constant: Decomposition or Not?

In summary, when integrating the fraction 1/(r(u-r)) where u is a constant, it is helpful to decompose the fraction using partial fractions. This will result in the equation 1/(ru) + 1/(u(u-r)). Integrating these terms will give the solution ln(r/u-r)/u. However, this solution may differ from others that include r0, which may need to be specified in the problem.
  • #1
syj
55
0

Homework Statement



i need to integrate:
[itex]\frac{1}{r(u-r)}dr[/itex]

Homework Equations



u is a constant

The Attempt at a Solution



im not sure if i should decompose the fraction. i tried that, but it didnt seem to be of any help.
 
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  • #2
Yes, use partial fractions. Show us how it didn't help. It should have.
 
  • #3
ok
so i get
[itex]\frac{1}{r(u-r)}=\frac{A}{r}+\frac{B}{u-r}[/itex]

so that
[itex]B-A=0[/itex]
and
[itex]Au=1[/itex]

so i got
[itex]A=\frac{1}{u}=B[/itex]

this gives me
[itex]\frac{1}{ur}+\frac{1}{u(u-r)}[/itex]
 
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  • #4
Good! Now, what do you get when you intgrate those?
 
  • #5
[itex]\int(\frac{1}{ur}+\frac{1}{u(u-r)})[/itex]

[itex]=\frac{1}{u}\int\frac{1}{r}+\int\frac{1}{u-r}[/itex]

[itex]=\frac{1}{u}(ln(r)-ln(u-r))[/itex]

[itex]=\frac{1}{u}(ln\frac{r}{u-r})[/itex]

my problem is that i don't know where to put [itex]r_0[/itex] here. The solution has [itex]r_0[/itex]
 
  • #6
Then perhaps you should have told us from the start what the problem really is. The problem you posted does not have any r0.
 
  • #7
oh, sorry what i mean is that when it integrates they have [itex]r_0[/itex], sorry, i meant the solution has [itex]r_0[/itex]
 

1. What is integration of a fraction?

Integration of a fraction is the process of finding the antiderivative of a fraction, or the function whose derivative is equal to the given fraction.

2. Why is integration of a fraction important?

Integration of a fraction is important because it allows us to solve real-world problems involving rates of change, such as finding the total distance traveled given a velocity function.

3. How do you integrate a fraction?

To integrate a fraction, we use the power rule, which states that the integral of x^n is (x^(n+1))/(n+1), where n is any real number except for -1. We also use other integration techniques such as substitution and integration by parts.

4. Can all fractions be integrated?

No, not all fractions can be integrated. Some fractions have complex integrals that cannot be expressed in terms of elementary functions. These are known as improper integrals.

5. How is integration of a fraction related to differentiation?

Integration and differentiation are inverse operations. This means that the integral of a function is the antiderivative of that function, and the derivative of an antiderivative is the original function. So, integration of a fraction is related to differentiation in that it is the reverse process of finding the derivative of a fraction.

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