An Integral With A Square Root In The Denominator

In summary, the conversation suggests finding a closed form solution for the given integral and potentially using partial fractions or integration by parts. It is also mentioned that some manipulation with trigonometric identities may be necessary.
  • #1
Radek Vavra
4
0
How would you integrate it?

[itex]\int \frac{d \varphi}{\sqrt{1 + \frac{a^2 b^2 \sin^2 \alpha}{(a \sin \varphi + b \sin (\alpha - \varphi))^2}}}[/itex]

I know that solving it numerically would probably be easier, but I would prefer a closed form solution in this case.
 
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  • #2
Try putting it as ^-.5 and also remember that a, b and sin^2(alpha) are all constants with respect to d(phi).
With that in mind try using partial fractions and/or By parts. Tell if You solve it!

Also pay attention to the fact that every term in the square root is a perfect square!
 
  • #3
It is a little work, but as a first step: asinφ + bsin(α-φ) = csin(φ+β) where c and β are constants depending on a,b, and α.
 

What is an integral with a square root in the denominator?

An integral with a square root in the denominator is a type of mathematical expression that involves solving for the area under a curve where the integrand (function being integrated) has a square root in the denominator. This type of integral is also known as a rational integral.

What is the process for solving an integral with a square root in the denominator?

The process for solving an integral with a square root in the denominator involves using a technique called integration by substitution. This involves substituting a new variable for the expression under the square root and then using standard integration techniques to solve the resulting integral.

What are some common examples of integrals with a square root in the denominator?

Some common examples of integrals with a square root in the denominator include the arc length of a circle, the area under a parabola, and the volume of a sphere. These types of integrals are commonly used in physics, engineering, and other fields to solve real-world problems.

Are there any special cases or techniques for solving integrals with a square root in the denominator?

Yes, there are some special cases and techniques that can be used for solving integrals with a square root in the denominator. For example, if the integrand is a rational function (a polynomial divided by a polynomial), then the integral can be solved using partial fractions. Additionally, trigonometric substitutions can be used in some cases to simplify the integral.

Why are integrals with a square root in the denominator important in mathematics?

Integrals with a square root in the denominator are important in mathematics because they allow us to calculate important quantities such as areas, volumes, and arc lengths. They also provide a deeper understanding of the relationship between a function and its integral, and they are essential in solving real-world problems in various fields of study.

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