Math Problem Statement
Beta functions integral from 0 to infinity x²/(1+x)^6=1/30 prove that
Solution
To prove that the integral
we'll show how it can be evaluated using the Beta function.
Step 1: Rewrite the Integral
We start by rewriting the integrand in a form that suggests a substitution related to the Beta function.
Consider the integral:
Now, let's express the integrand as:
Step 2: Substitution to Convert to Beta Function
Let us use the substitution:
With this substitution:
So, the integral becomes:
Simplify the expression:
Step 3: Recognize the Beta Function
The integral is the Beta function . Here, we identify:
Thus, the integral is:
Step 4: Use the Beta Function Property
We know that the Beta function is related to the Gamma function by the identity:
For , we have:
Using the fact that for a positive integer :
Therefore:
Conclusion
We have shown that:
This completes the proof.
Would you like to explore any of these concepts further? Here are 5 related questions to consider:
- What are the properties and applications of the Beta function in various fields?
- How is the substitution method useful in solving integrals related to Beta and Gamma functions?
- How do the Beta and Gamma functions relate to combinatorial identities?
- Can you derive the integral using a different substitution method?
- What are some other important integrals that involve the Beta function?
Tip: When evaluating integrals, recognizing forms that match Beta or Gamma functions can simplify your work significantly.
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Math Problem Analysis
Mathematical Concepts
Integral Calculus
Beta Function
Gamma Function
Formulas
Beta function definition
Gamma function identity
Theorems
Beta-Gamma relationship
Suitable Grade Level
Advanced Mathematics