Integrating Using Partial Fractions

In summary, the conversation is about solving an arc length problem in three dimensions using the vector r(t) = <et, 1, t> from t=0 to t=1. The solution involves taking the derivative and simplifying it until it becomes ∫√(e2t+1)dt, which can then be further simplified using partial fractions. The speaker encountered some difficulties with the process, but eventually found a solution by adding and subtracting terms.
  • #1
Jay9313
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0

Homework Statement


This is an arc length problem in three dimensions. I was given the vector r(t)=<et, 1, t> from t=0 to t=1


Homework Equations


Arc Length= [itex]\int[/itex] |[itex]\sqrt{r'(t)}[/itex]| dt from t1 to t2
where |[itex]\sqrt{r'(t)}[/itex]| is the magnitude of the derivative of the vector

The Attempt at a Solution



I took the derivative and got the magnitude and simplified it down to
∫ √(e2t+1) dt
I then set u=e2t+1
I then simplified and substituted until I got to:∫[itex]\frac{\sqrt{(u)}}{u-1}[/itex] du

My professor said to use partial fractions from here, but I'm not sure how to do that.
 
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  • #2
Try one more substitution to get rid of the square root on top first. Then do partial fractions.
 
  • #3
That's not really working at all
 
  • #4
What did you try?
 
  • #5
I split it up, and it made it worse, and I had to do long division, but I got an answer. It's nasty, but I got an answer.
 
  • #6
Presumably, you ended up with something like ##\frac{v^2}{v^2-1}## after the substitution. A good technique to avoid doing long division is to add and subtract judiciously:
$$\frac{v^2}{v^2-1} = \frac{(v^2-1)+1}{v^2-1} = 1 + \frac{1}{v^2-1}.$$
 

1. What is partial fractions integration?

Partial fractions integration is a method used to decompose a rational function into simpler fractions, making it easier to integrate. This method is especially useful when dealing with integrals involving polynomials.

2. When should partial fractions integration be used?

Partial fractions integration is typically used when the integrand (the function being integrated) is a rational function, meaning it is a ratio of two polynomials. This method can also be used when the degree of the numerator is smaller than the degree of the denominator.

3. How is partial fractions integration done?

Partial fractions integration involves breaking down the rational function into simpler fractions using a process called partial fraction decomposition. This can be done by expressing the rational function as a sum of simpler fractions with unknown coefficients, and then solving for those coefficients.

4. What are the benefits of using partial fractions integration?

Partial fractions integration can simplify complex integrals and make them easier to solve. It also allows for the use of basic integration rules, making the integration process more straightforward and efficient.

5. Are there any limitations to using partial fractions integration?

Partial fractions integration can only be used when the integrand is a rational function. It also requires the denominator to be factorable into linear and quadratic terms. If the denominator cannot be factored, other integration methods may need to be used.

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