Solving Integrals Using the Beta and Gamma Functions

Introduction

Integration is a fundamental concept in calculus, and certain integrals can be solved more efficiently by employing special functions such as the Beta and Gamma functions. In this article, we will demonstrate how to solve the integral ( I int x^{20} (2 - x^{5}) , dx ) using these advanced functions.

Step 1: Rewrite the Integrand

Before solving the integral, we first rewrite the integrand in a more manageable form. The integrand is given by:

( x^{20} (2 - x^{5}) )

We can simplify this expression by recognizing a more convenient way to represent it. However, in this specific case, it is already in a form that can be directly integrated using the Beta function after some manipulation.

Step 2: Change of Variables

To simplify the integral, we perform a change of variables. Let us use the substitution:

( x 2u )

which implies:

( dx 2 , du )

Keeping the integral general, we get:

( I int 2u^{20} (2 - 2u^{5}) 2 , du ) (2^{21} int u^{20} (1 - u^{5}) , du )

Step 3: Recognizing the Beta Function

The integral ( int u^{20} (1 - u^{5}) , du ) can be recognized as a Beta function. The Beta function, ( B(x, y) ), is defined as:

( B(x, y) int_{0}^{1} t^{x-1} (1 - t^{y-1}) , dt )

By making a substitution and adjusting the limits, we can write:

( int_{0}^{1} u^{20} (1 - u^{5}) , du B(21, 6) )

Step 4: Calculate the Beta Function

The Beta function can be expressed in terms of the Gamma function:

( B(x, y) frac{Gamma(x) Gamma(y)}{Gamma(x y)} )

For our specific case:

( B(21, 6) frac{Gamma(21) Gamma(6)}{Gamma(27)} )

Using properties of the Gamma function, we know:

( Gamma(21) 20! ), ( Gamma(6) 5! ), and ( Gamma(27) 26! )

Thus, we have:

( B(21, 6) frac{20! cdot 5!}{26!} )

Step 5: Substitute Back

Substituting back into our expression for ( I ), we obtain:

( I 2^{21} B(21, 6) 2^{21} cdot frac{20! cdot 5!}{26!} )

Final Result

Therefore, the integral evaluates to:

( I 2^{21} cdot frac{20! cdot 5!}{26!} C )

where ( C ) is the constant of integration. This completes the solution using the Gamma and Beta functions.

Further Information

The Beta function was initially studied by the famous mathematician Leonhard Euler, and is also known as the Euler integral of the first kind. It was named by the mathematician Jacques Binet. The Beta function is represented by the uppercase letter ( B ), which is the Greek capital of beta.

The Beta function is a special function that can be expressed in the form of an integral:

( B(x, y) int_{0}^{1} t^{x-1} (1 - t^{y-1}) , dt )

where ( x > 0 ) and ( y > 0 ).

For more information, you can watch the following video: