Integral of product of irrational powers

In summary, the conversation discusses the integration of irrational "polynomials" using dimensional regularization. Mathematica is able to solve the indefinite integral and provides a formula for the solution. The individual is also interested in finding reference materials on integrating irrational powers.
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niterida
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Hello,

I'm trying to integrate a product of irrational "polynomials" that have arisen through dimensional regularization. As a warmup, I'm trying to understand how to integrate:

(w-a)^b (w-c)^d,

where b and d are arbitrary numbers, possibly irrational.

Mathematica can solve this indefinite integral, getting:

(((a - w)/(a - c))^-b (-a + w)^b (-c + w)^(1 + d)
Hypergeometric2F1[-b, 1 + d, 2 + d, (-c + w)/(a - c)])/(1 + d)

Can anyone explain how the answer is obtained?

Also, if there are any good tables (in print or online) or references on the subject of integrating
irrational powers, I would be very interested. Thank you.
 
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FAQ: Integral of product of irrational powers

What is the definition of "integral of product of irrational powers"?

The integral of product of irrational powers refers to the process of finding the antiderivative or indefinite integral of a function that is a product of irrational powers, such as √x or xπ.

Why is it important to know how to integrate products of irrational powers?

Knowing how to integrate products of irrational powers is important in various fields of science, such as physics and engineering. It allows us to solve complex problems involving these types of functions and make accurate predictions and calculations.

What are some common techniques used to solve integrals of product of irrational powers?

Some common techniques used to solve integrals of product of irrational powers include substitution, integration by parts, and partial fractions. These techniques involve manipulating the given function in a way that makes it easier to integrate.

Are there any special cases when integrating products of irrational powers?

Yes, there are special cases when integrating products of irrational powers. For example, if the function contains a radical and a polynomial, we can use a trigonometric substitution to simplify the integral. Additionally, if the function contains a square root and a polynomial, we can use the method of completing the square.

What are some real-world applications of integrals of product of irrational powers?

Integrals of product of irrational powers have various real-world applications, such as calculating the center of mass of an object with a varying density, determining the volume of irregular shapes, and finding the work done by a variable force over a given distance.

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