Integral of $\frac{\sin^3\theta}{(c+br\cos\theta+(\sin^2\theta) r^2)^3}$

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In summary, the conversation discusses a specific integral and various strategies for solving it. The suggested method involves completing the square and making a change of variables from theta to cos(theta), followed by partial fractions decomposition and integration with respect to r.
  • #1
quasar987
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Does anyone know how to do this integral? Mapple just thinks and think and never gives an answer. I couldn't find a primitive in a table either.

[tex]\int_{0}^{\pi}\int_{0}^{+\infty}\frac{r^4\sin^3\theta}{(c+br\cos\theta+(\sin^2\theta )r^2)^3}drd\theta[/tex]

In one attempt, I used a table to reduce the r integral to

[tex]\int_{0}^{\pi}\int_0^{+\infty}\frac{\sin^3\theta}{(c+br\cos\theta+(\sin^2\theta) r^2)^3}dr[/tex]

But what's the integral of that? :confused:
 
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  • #2
Complete the square within the denominator with rcos(theta) as one of your terms within that square.
Then, make a change of variables from theta to cos(theta) (i.e, integrate from 1 to -1); probably, you need to use partial fractions decomposition.

You'll end up (hopefully!) with a rational function in r, which you then are to integrate with respect to r.
 
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Related to Integral of $\frac{\sin^3\theta}{(c+br\cos\theta+(\sin^2\theta) r^2)^3}$

1. What is the meaning of the integral in the given equation?

The integral represents the area under the curve of the given function, which is a measure of the total effect of the function. In this case, it represents the total effect of the sine function raised to the third power over the given expression.

2. How can one solve the integral in the given equation?

The integral can be solved using various techniques, such as substitution, integration by parts, or using trigonometric identities. It is important to carefully identify the appropriate technique to use in order to solve the integral.

3. What are the practical applications of this integral?

This integral has various applications in physics, engineering, and mathematics. It is commonly used to calculate the work done by a varying force, the center of mass of a distribution, or the moment of inertia of a rotating object.

4. Can this integral be solved analytically?

Yes, this integral can be solved analytically using the aforementioned techniques. However, in some cases, it may not have a closed form solution and numerical methods may be required to approximate the value.

5. Are there any special cases or restrictions for this integral?

Yes, the integral may have different solutions depending on the values of the constants c and b. Additionally, the limits of integration and the range of the variable theta may also affect the solution. It is important to carefully consider these factors when solving the integral.

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