Proving Trigonometric Integrals: A Detailed Look at the Equality of Two Definite Integrals

Understanding and proving trigonometric identities is a fundamental aspect of higher mathematics. In this article, we will explore a specific identity and provide a detailed step-by-step process to prove its correctness. This process will involve various mathematical techniques, including trigonometric substitutions and integral manipulation.

The Problem Statement

The following identity is always true for any a and any function f(x):

[ int_{a}^{frac{pi}{2} - a} frac{f(x)sin{x}}{f(x)sin{x}f(x)cos{x}}dx int_{a}^{frac{pi}{2} - a} frac{f(x)cos{x}}{f(x)sin{x}f(x)cos{x}}dx ]

This identity has particular significance for specific functions, such as f(x) x^3. In this article, we will demonstrate step-by-step how to prove the equality of these two integrals, focusing on the case where f(x) x^3.

Proving the Identity

To prove the given identity, we will follow a detailed approach, breaking down the process into manageable steps. This will make it easier to understand and follow the proof.

Step 1: Simplifying the Integrands

First, we simplify the denominators of both integrands. Let's start by factoring out fsin(x)f(x)cos(x) in the left-hand side:

[ int_{a}^{frac{pi}{2} - a} frac{f(x)sin{x}}{f(x)sin{x}f(x)cos{x}} dx int_{a}^{frac{pi}{2} - a} frac{1}{f(x)cos{x}} dx ]

Similarly, in the right-hand side, we factor out f(x)sin(x)f(x)cos(x) and simplify:

[ int_{a}^{frac{pi}{2} - a} frac{f(x)cos{x}}{f(x)sin{x}f(x)cos{x}} dx int_{a}^{frac{pi}{2} - a} frac{1}{f(x)sin{x}} dx ]

Step 2: Choosing a Function

For this specific problem, we choose the function f(x) x^3. This simplifies the integrands to:

[ int_{a}^{frac{pi}{2} - a} frac{1}{cos{x}} dx and int_{a}^{frac{pi}{2} - a} frac{1}{sin{x}} dx ]

These integrals are referred to as the secant and cosecant functions, respectively.

Step 3: Performing Trigonometric Substitutions

Now, we will prove the identity by transforming the left-hand side integral. We divide the top and bottom of the integrand by (cos^3(x)) and the right-hand side by (sin^3(x)):

[ frac{sin^3(x)}{sin^3(x) cos^3(x)} frac{tan^3(x)}{tan^3(x) - 1} ]

[ frac{cos^3(x)}{sin^3(x) cos^3(x)} frac{cot^3(x)}{1 - cot^3(x)} ]

Step 4: Applying a Substitution

To further simplify, we make the substitution (y frac{pi}{2} - x). This changes the variable of integration and must account for the new bounds. Since (dy -dx) and the bounds reverse, we have:

[ int_0^{frac{pi}{2}} frac{cot^3(x)}{1 - cot^3(x)} dx -int_{frac{pi}{2}}^{0} frac{cot^3(y)}{1 - cot^3(y)} dy ]

Reversing the limits of integration changes the sign, simplifying the integral to:

[ int_0^{frac{pi}{2}} frac{tan^3(y)}{tan^3(y) - 1} dy ]

Since the variable of integration is purely symbolic, we can equate the two integrals:

[ int_0^{frac{pi}{2}} frac{cos^3(x)}{sin^3(x) cos^3(x)} dx int_0^{frac{pi}{2}} frac{tan^3(x)}{tan^3(x) - 1} dx ]

Conclusion

Thus, we have demonstrated the equality of the given integrals:

[ int_0^{frac{pi}{2}} frac{sin^3(x)}{sin^3(x) cos^3(x)} dx int_0^{frac{pi}{2}} frac{tan^3(x)}{tan^3(x) - 1} dx int_0^{frac{pi}{2}} frac{cos^3(x)}{sin^3(x) cos^3(x)} dx ]

This proof showcases the power of trigonometric identities and integral manipulation in proving mathematical statements. Understanding these techniques is crucial for further studies in calculus and mathematical proofs.