Improper Integral Help: Solving \int\frac{1}{\sqrt[3]{x-1}}

In summary, the problem is to solve the integral \int\frac{1}{\sqrt[3]{x-1}} with upper limit of integration 1 and lower limit 0. The antiderivative of the integral is \frac{3}{2}(x-1)^(2/3) + C, and to find the solution, the limit of the integral \lim_{b \to 1^-} \int_0^b \frac{dx}{\sqrt[3]{x-1}} must be taken. The limit approaches 1, as f(x) does not exist. The constant of integration is not needed for a definite integral.
  • #1

Homework Statement



Solve the integral [tex]\int[/tex][tex]\frac{1}{\sqrt[3]{x-1}}[/tex]. Upper limit of integration is 1 while lower limit is 0.

Homework Equations



N/A.

The Attempt at a Solution



The only thing that I'm sure about is that the antiderivative of the integral is [tex]\frac{3}{2}[/tex](x-1)^(2/3) + C. I know that i need to take the limit of the integral, but I am not sure what the limit should be approaching. 1 Maybe? Since f(x) DNE there. Since I'm stuck on that, I'm kinda stuck on everything else besides the antiderivative.

Any help would be great! Thanks.
 
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  • #2
This is what you want:
[tex]\lim_{b \to 1^-} \int_0^b \frac{dx}{\sqrt[3]{x-1}}[/tex]

For a definite integral you don't need the constant of integration.
 

1. What is an improper integral?

An improper integral is an integral where one or both of the limits of integration are infinite or the integrand function has a vertical asymptote in the interval of integration.

2. How is an improper integral different from a regular integral?

A regular integral has finite limits of integration and the integrand function is continuous in the interval of integration. An improper integral has infinite limits of integration or a discontinuous integrand function, making it more difficult to evaluate.

3. What is the significance of an improper integral in mathematics?

Improper integrals are important in mathematics because they allow us to evaluate areas or volumes of regions that cannot be represented by regular integrals. They also have applications in physics, engineering, and other scientific fields.

4. How do you determine if an improper integral is convergent or divergent?

To determine the convergence or divergence of an improper integral, you can use various methods such as the comparison test, limit comparison test, or the integral test. These methods involve evaluating the limit of the integral as the limits of integration approach infinity or a point of discontinuity.

5. Can all improper integrals be evaluated analytically?

No, not all improper integrals can be evaluated analytically. Some integrals may require more advanced techniques such as integration by parts or substitution, and others may not have a closed-form solution and can only be approximated numerically.

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