Solving Improper Integral: \int \frac{dx}{(ax+b)^2}

In summary, an improper integral is an integral with either infinite limits of integration or a vertical asymptote in the interval, requiring special techniques to solve. These techniques include rewriting the integral as a limit, splitting it into smaller intervals, using substitutions or known formulas, and solving for real-world quantities. For the specific case of \int \frac{dx}{(ax+b)^2}, the substitution method can be used. Special cases and restrictions to keep in mind when solving improper integrals include finite limits, continuous integrands, absence of oscillations or infinite discontinuities, and non-divergent values within the interval.
  • #1
suspenc3
402
0
Hi, for some reason I can't remember and I've been looking everywhere for some info but can't find anything. I am trying to find out the answer to :

[tex] \int_1^i^n^f^i^n^i^t^e \frac{1}{(3x+1)^2}dx[/tex]

what is the integral of [tex] \int \frac{dx}{(ax+b)^2}[/tex]?

Thanks
 
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  • #2
Try the substitution u=ax+b.
 
  • #3
ahhh, got it, THanks
 

1. What is an improper integral?

An improper integral is a type of integral where either the limits of integration are infinite or the integrand has a vertical asymptote within the limits of integration. In these cases, the integral does not have a finite value and requires special techniques to solve.

2. How do you solve an improper integral?

To solve an improper integral, you can use one of the following methods:
- Rewrite the integral as a limit of a definite integral and evaluate the limit
- Split the integral into smaller intervals and evaluate each interval separately
- Use a substitution or change of variable to transform the integral into a form that can be easily evaluated
- Use a known formula or technique for solving specific types of improper integrals, such as the comparison test or the Cauchy principal value.

3. What is the purpose of solving an improper integral?

The purpose of solving an improper integral is to find the exact or approximate value of an integral that would otherwise be undefined. Improper integrals are often used in physics and engineering to model real-world situations and calculate important quantities such as areas, volumes, and probabilities.

4. How do you solve the improper integral \int \frac{dx}{(ax+b)^2}?

This specific improper integral can be solved by using the substitution method. Let u = ax+b, then du = a dx, and the integral becomes \int \frac{du}{au^2}. This integral can then be evaluated using the power rule for integrals, and the resulting antiderivative can be substituted back in terms of x.

5. Are there any special cases or restrictions when solving improper integrals?

Yes, there are some special cases and restrictions to keep in mind when solving improper integrals. These include:
- The integral must have a finite limit as the variable approaches the limit of integration
- The integrand must be continuous in the interval of integration
- The integral must not be oscillatory or have an infinite number of discontinuities
- The integral must not be divergent or have infinite values within the interval of integration.

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