1. Introduction to the Integral Evaluation

When dealing with integrals that involve powers of sine and cosine, a common problem in calculus and mathematical analysis, it is essential to employ efficient techniques to simplify and evaluate these expressions. This article delves into the evaluation of the integral ( ∫_{0}^{frac{pi}{2}} sin^n x cos^n x dx ), providing a detailed step-by-step approach that highlights the use of several key mathematical tools, such as the beta function, gamma function, and Legendre's duplication formula.

2. Initial Simplification Using Symmetry

The integral in question is:

( ∫_{0}^{frac{pi}{2}} sin^n x cos^n x dx )

We start by using the integral property:

( ∫_{0}^{2a} f(x) dx 2 ∫_{0}^{a} f(x) dx ), whenever ( f(x) f(2a - x) ).

In our case, let ( a frac{pi}{4} ), and we have:

( ∫_{0}^{frac{pi}{2}} sin^n x cos^n x dx 2 ∫_{0}^{frac{pi}{4}} sin^n x cos^n x dx ).

3. Substitution and Beta Function

To further simplify, we make the substitution:

( sin 2x sqrt{t} implies dx frac{dt}{4 sqrt{t} sqrt{1-t}} ).

This transforms the integral into:

( frac{1}{2^{n-1}} ∫_{0}^{1} t^{frac{n}{2} - frac{1}{2}} (1-t)^{-frac{1}{2}} dt )

Recognizing the form of the beta function:

( B(a, b) frac{Γ(a) Γ(b)}{Γ(a b)} ), we can rewrite the integral as:

( frac{1}{2^{n-1}} B left( frac{n 1}{2}, frac{1}{2} right) )

4. Applying Legendre’s Duplication Formula

Recall Legendre's duplication formula:

( Γ(z frac{1}{2}) frac{sqrt{π} Γ(2z)}{2^{2z-1} Γ(z)} )

Using ( z frac{n}{2} ) in the duplication formula, we get:

( Γ left( frac{n 1}{2} right) frac{sqrt{π} Γ(n)}{2^{n-1} Γ left( frac{n}{2} right)} )

This simplifies the original integral to:

( ∫_{0}^{frac{pi}{2}} sin^n x cos^n x dx frac{sqrt{π} Γ left( frac{n 1}{2} right)}{2^{n-1} Γ left( frac{n}{2} 1 right)} )

Further simplification using the duplication formula yields:

( ∫_{0}^{frac{pi}{2}} sin^n x cos^n x dx frac{π Γ(n)}{4^n left( frac{n}{2} 1 right) Γ left( frac{n}{2} right)^2} )

5. Special Cases for Even and Odd Positive Integers

Let's consider the integral for even and odd positive integers:

(a) For Even Positive Integer: Write ( n 2k ), where ( k ∈ N ).

These integral properties:

( Γ(z) z Γ(z-1) (z-1)! ), yield:

( ∫_{0}^{frac{π}{2}} sin^{2k} x cos^{2k} x dx frac{π 2k-1!}{16^k k(k-1)!^2} )

(b) For Odd Positive Integer: Write ( n 2k-1 ), where ( k ∈ N ).

The integral simplifies to:

( ∫_{0}^{frac{π}{2}} sin^{2k-1} x cos^{2k-1} x dx frac{2k-2!}{4^{2k-1} left( k-frac{1}{2} right) left( k-frac{3}{2} right) cdots frac{3}{2} } )

6. Conclusion

In conclusion, evaluating integrals of sine and cosine to the power of ( n ) can be approached through a combination of symmetry properties, substitutions, and special mathematical formulas. These methods not only simplify the integrals but also provide insights into their underlying mathematical structure.