# Solving Definite Integral with a, b Parameters

• zeebek
In summary, the conversation discusses a definite integral involving a and b as positive numbers. The attempt at a solution involves trying to integrate by parts and differentiate by parameter a, but is unsuccessful due to the square root. The conversation also mentions the possibility of representing the integral using generalized functions, but this is deemed impossible due to the values of a and b.
zeebek
This is not really a homework.

## Homework Statement

I am trying to solve one definite integral

## Homework Equations

$$\int_0^{2 \pi} \frac{\sin^2{t}}{\sqrt{a\cos{t} + b}} dt$$

where $$a, b$$ are some positive numbers.

## The Attempt at a Solution

I tried integrate by parts, also differentiate by parameter $$a$$. Does not help because of the square root. In this case I got some diff.equation for the integral I want, but I need to solve even more nasty integrals:

either

$$\int_0^{2 \pi} \sqrt{a\cos{t} + b} \cos{t} dt$$

or

$$\int_0^{2 \pi} \frac{1}{\sqrt{a\cos{t} + b}} dt$$

You joking, right? Even by breaking this up into two separate integrals, the first one 1/sqrt(a*cos(t)+b), t=0..2pi and second one -cos^2(t)/sqrt(a*cos(t)+b), t=0..2pi.. This gives some complete elliptic integral of first one for first part, and series expansion for second part. At best you can get a complete series expansion of this with http://www.wolframalpha.com/input/?i=Expand+(1-cos^2(t))/sqrt(a*cos(t)+b)" and numerically evaluate it if you know a and b.

Last edited by a moderator:
cronxeh said:
You joking, right? Even by breaking this up into two separate integrals, the first one 1/sqrt(a*cos(t)+b), t=0..2pi and second one -cos^2(t)/sqrt(a*cos(t)+b), t=0..2pi.. This gives some complete elliptic integral of first one for first part, and series expansion for second part. At best you can get a complete series expansion of this with http://www.wolframalpha.com/input/?i=Expand+(1-cos^2(t))/sqrt(a*cos(t)+b)" and numerically evaluate it if you know a and b.

Thank you for your reply. As I said this is not a homework problem. In fact if somebody could help me represent this intergral via some generalized functions this would do. Sorry for the misconception.

Last edited by a moderator:
zeebek said:
Thank you for your reply. As I said this is not a homework problem. In fact if somebody could help me represent this intergral via some generalized functions this would do. Sorry for the misconception.

I can't see how this could be done. a and b make this impossible

## 1. How do I solve a definite integral with parameters a and b?

Solving a definite integral with parameters a and b requires using the fundamental theorem of calculus and the substitution method. First, evaluate the indefinite integral of the function with respect to the variable. Then, substitute the values of a and b and subtract the lower limit from the upper limit to get the final answer.

## 2. Can I solve a definite integral with parameters using integration by parts?

Yes, you can solve a definite integral with parameters using integration by parts. This method involves choosing two parts of the integrand and applying the integration by parts formula to solve for the integral. However, it may not always be the most efficient method and may require multiple iterations.

## 3. What is the importance of using parameters in definite integrals?

Parameters in definite integrals allow for more flexibility in solving for the integral. They allow you to solve for a range of values rather than just a specific number. Additionally, using parameters can simplify the integrand and make it easier to solve.

## 4. How do I know when to use substitution or integration by parts when solving a definite integral with parameters?

The choice between using substitution or integration by parts when solving a definite integral with parameters depends on the complexity of the integrand. If the integrand contains a product of two functions, integration by parts may be more appropriate. If the integrand contains a nested function, substitution may be a better choice. It is helpful to try both methods and see which one yields a simpler solution.

## 5. Is there a general method for solving definite integrals with parameters?

There is no one general method for solving definite integrals with parameters. It often depends on the specific integral and the techniques used may vary. It is important to have a good understanding of various integration methods such as substitution, integration by parts, and partial fractions to effectively solve definite integrals with parameters.

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