Solving Integrals with Special Functions: The Case of ( int frac{dx}{sqrt{ax^n}} )

When dealing with integrals that do not have a solution in terms of elementary or standard functions, we often turn to special functions or symbolic integration techniques. This article explores how to solve the integral ( int frac{dx}{sqrt{ax^n}} ) using these methods.

Introduction to Special Functions

A special function is a mathematical function that has established names and notations due to their importance in mathematical analysis, functional analysis, physics, or other applications. One common special function encountered in the solution of integrals is the hypergeometric function, denoted as ( _2F_1(a, b, c, z) ), and the elliptic integral of the first kind, denoted as ( F(phi, m) ).

Solving the Integral ( int frac{1}{sqrt{7x^3}} dx )

Consider the integral:

[ int frac{1}{sqrt{7x^3}} dx ]

Using a Computer Algebra System (CAS) like Mathematica, we can solve this integral symbolically. The solution involves the hypergeometric function ( _2F_1(a, b, c, z) ). Let's delve into the steps to reach this solution.

Mathematica Solution

To solve this integral using Mathematica, we type the following command:

ExpToTrig[Integrate[1/Sqrt[x^3 7] x]]

The result is:

[ frac{1}{sqrt[4]{3} sqrt{7x^3-7}} cdot left(2 sqrt[3]{7} left(frac{sqrt{3}}{2} frac{i}{2} right) cdot sqrt{-frac{i x}{2 sqrt[3]{7}} - frac{sqrt{3} x}{2 sqrt[3]{7}} - frac{i}{2} frac{sqrt{3}}{2}} cdot sqrt{frac{i sqrt{3} x^2}{2 7^{2/3}} - frac{x^2}{2 7^{2/3}} left(frac{i sqrt{3} x}{2 sqrt[3]{7}} frac{x}{2 sqrt[3]{7}}right) 1} cdot Fleft(text{In}^{-1} left(frac{sqrt{-frac{i x}{2 sqrt[3]{7}} - frac{sqrt{3} x}{2 sqrt[3]{7}} - frac{i}{2} frac{sqrt{3}}{2}}}{sqrt[4]{3}} right)right) right) C ]

Where ( F(phi, m) ) is the elliptic integral of the first kind:

[ F(phi, m) int_0^{phi} left(1 - m sin^2 theta right)^{-frac{1}{2}} dtheta ]

Verification with Arbitrary Power and Number

To verify the solution, consider the general form of the integral:

[ int frac{1}{sqrt{ax^n}} dx ]

The solution is given by:

[ int frac{1}{sqrt{7x^3}} dx frac{x cdot _2F_1 left(frac{1}{3}, frac{1}{2}, frac{4}{3}, -frac{x^3}{7} right)}{sqrt{7}} C ]

This solution also satisfies the integral's series expansion around ( x 0 ). Comparing the series expansion of ( frac{1}{sqrt{7x^3}} ) and the integral with the hypergeometric function, we can verify the equivalence.

Generalized Result for Arbitrary Index

The generalized result for the integral with an arbitrary index is:

[ int frac{1}{(ax^n)^{1/p}} dx int frac{1}{sqrt[p]{ax^n}} dx x (ax^n)^{-1/p} left(frac{x^n}{a} - 1right)^{1/p} cdot _2F_1 left(frac{1}{n}, frac{1}{p}, 1, frac{x^n}{a} - 1right) C ]

where ( _2F_1(a, b, c, z) ) is the hypergeometric function.

Conclusion

Understanding and solving integrals that involve special functions like the hypergeometric function and elliptic integrals is crucial in advanced mathematics and applications in physics and engineering. Using tools like Mathematica can provide detailed symbolic solutions to these complex problems, allowing for a deeper and more complete understanding of integral calculus.