Need help with this definite integral

In summary, a definite integral is a mathematical concept used to find the area under a curve or the accumulation of a function over a specific interval. To solve a definite integral, you need to first identify the function and the interval, use integration techniques, and evaluate the integral by plugging in the upper and lower limits. The main difference between a definite and indefinite integral is that a definite integral has a defined interval, while an indefinite integral does not. You can use a calculator to solve a definite integral, but it is recommended to understand the concept and solve it by hand. Some common tips for solving definite integrals include identifying the function and interval correctly, using appropriate techniques, checking for symmetry, and practicing regularly.
  • #1
3
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I'm having a tough time with this integral:

$$\int_{0}^\infty \frac{x^2 \, dx}{x^4+(a^2+\frac{1}{b^2})x^2+\frac{2a^2}{b^2}}$$
where $$a, b \in \Bbb R^+$$ I tried using the residue theorem, but the roots of the denominator I found are quite complicated, and I got stuck.

What contour should I use? Is there an alternative method? I would appreciate any advice.
 
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  • #2
There are no singularities in the region of integration. This looks like partial fractions might be the proper approach.
 
  • #3
For partial fractions, the denominator terms will be in terms of [itex]x^2\ not\ x[/itex].
 

1. What is a definite integral?

A definite integral is a mathematical concept used to find the area under a curve or the accumulation of a function over a specific interval. It is represented by the symbol ∫ and has a lower and upper limit that defines the interval over which the function is being integrated.

2. How do I solve a definite integral?

To solve a definite integral, you need to first identify the function and the interval over which you want to integrate. Then, you can use various integration techniques such as the power rule, substitution, or integration by parts to find the antiderivative of the function. Finally, you can evaluate the definite integral by plugging in the upper and lower limits into the antiderivative.

3. What is the difference between a definite and indefinite integral?

The main difference between a definite and indefinite integral is that a definite integral has a defined interval over which the function is being integrated, while an indefinite integral does not. An indefinite integral results in a general antiderivative of a function, while a definite integral gives a specific numerical value.

4. Can I use a calculator to solve a definite integral?

Yes, you can use a calculator to solve a definite integral. Most scientific and graphing calculators have built-in integration functions that can quickly solve integrals using numerical methods. However, it is always recommended to understand the concept and solve integrals by hand to ensure accuracy.

5. What are some common tips for solving definite integrals?

Some common tips for solving definite integrals include identifying the function and interval correctly, using appropriate integration techniques, checking for symmetry and using it to simplify the integral, and practicing regularly to improve your integration skills.

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