Solving a Challenging Integral: Symmetry and Definite Integration Techniques
In this article, we will explore how to solve a challenging integral using a combination of symmetry techniques and definite integral properties. We will delve into the detailed steps and rational behind each transformation, ensuring that the problem is broken down into simpler, more manageable pieces.
Introduction to the Integral Problem
Consider the following integral:
[mathcal{I} int_{- pi}^{pi} frac{x sin x}{sqrt{1 x^2} sqrt{3 sin^2 x}} , dx]At first glance, this may seem daunting, but by applying some strategic steps, we can simplify and solve this integral effectively.
Initial Transformation and Symmetry Consideration
The first step is to consider the symmetry in the integral. We start by making the substitution (x rightarrow -x), which effectively changes the sign of the integrand:
[int_{- pi}^{pi} frac{x sin x}{sqrt{1 x^2} sqrt{3 sin^2 x}} , dx int_{- pi}^{pi} frac{- x sin x}{sqrt{1 x^2} sqrt{3 sin^2 x}} , dx]By adding the original integral and the transformed integral, we get:
[2mathcal{I} int_{- pi}^{pi} frac{x sin x}{sqrt{3 sin^2 x}} cdot left(frac{1}{sqrt{1 x^2}} frac{1}{sqrt{1 - x^2}}right) , dx]Notice that the terms inside the parentheses are equal to 1 when combined:
[frac{1}{sqrt{1 x^2}} frac{1}{sqrt{1 - x^2}} 1]This leads to:
[2mathcal{I} int_{- pi}^{pi} frac{x sin x}{sqrt{3 sin^2 x}} , dx]Simplification Using Symmetry and Properties of Even Functions
Since the resulting integral is an even function over a symmetric interval, we can simplify the integration range:
[mathcal{I} int_{0}^{pi} frac{x sin x}{sqrt{3 sin^2 x}} , dx]Next, we use another property of definite integrals to eliminate the (x) from the integrand:
[mathcal{I} frac{pi}{2} int_{0}^{pi} frac{sin x}{sqrt{3 sin^2 x}} , dx frac{pi}{2} int_{0}^{pi} frac{sin x}{sqrt{4 - cos^2 x}} , dx]Now, let's perform a substitution: (cos x t). Therefore, (-sin x , dx dt), and the limits of integration change from (0) to (pi) to (1) to (-1):
[frac{pi}{2} int_{-1}^{1} frac{-dt}{sqrt{4 - t^2}} -frac{pi}{2} int_{-1}^{1} frac{dt}{sqrt{4 - t^2}} frac{pi}{2} int_{-1}^{1} frac{dt}{sqrt{4 - t^2}}]Since the integrand is an even function, we can further simplify the integral:
[frac{pi}{2} int_{-1}^{1} frac{dt}{sqrt{4 - t^2}} pi int_{0}^{1} frac{dt}{sqrt{4 - t^2}}]Evaluating the Final Integral
The integral (int frac{dt}{sqrt{4 - t^2}}) is a standard form that can be evaluated using the arcsin function:
[pi int_{0}^{1} frac{dt}{sqrt{4 - t^2}} pi left[ arcsin left(frac{t}{2}right) right]_{0}^{1} pi left( arcsin left(frac{1}{2}right) - arcsin (0) right) pi left( frac{pi}{6} - 0 right) frac{pi^2}{6}]This simplifies to:
[2mathcal{I} frac{pi^2}{6}]Therefore, the value of the initial integral (mathcal{I}) is:
[mathcal{I} frac{pi^2}{12}]Conclusion
[boxed{int_{- pi}^{pi} frac{x sin x}{sqrt{1 x^2} sqrt{3 sin^2 x}} , dx frac{pi^2}{6}}]In conclusion, by leveraging the symmetry of the function and definite integral properties, we were able to solve this challenging integral effectively. This method showcases the power of symmetry and transformation in simplifying complex integrals.