Exploring the Divergence of the Integral of 2sin(2x)/sin(2x^4) Near x 0

Understanding the concept of an integral not converging, also known as divergence, is crucial in advanced mathematical analysis. This article delves into the specific case of the integral [ int_0^{pi/4} frac{2sin(2x)}{sin(2x^4)} , dx ], which is divergent due to its behavior near x 0.

Behavior Near x 0

The divergent nature of this integral near x 0 is closely tied to the behavior of the denominator, sin(2x^4), when x is close to zero. To analyze this, we rewrite the integral using the double angle identity:

[ int_0^{pi/4} frac{2sin(2x)}{sin(2x^4)} , dx ]

Since we are focused on the behavior near 0, we choose a small positive number alpha such that the integrand is both continuous and positive on the interval [0, alpha].

Divergence Analysis Using Direct Comparison Test

To prove the divergence of the integral over [0, alpha], we utilize the Direct Comparison Test. This test is particularly useful when dealing with integrands that are difficult to evaluate directly but can be compared to simpler functions whose integrals are known.

First, we use the classic inequality for sine:

[ frac{2}{pi}x leq sin{x} leq x quad text{for all } 0 leq x leq frac{pi}{2} ]

This geometrically implies that y sin{x} lies between y x (the tangent line at the origin) and y frac{2}{pi}x (the secant line connecting (0,0) to left(frac{pi}{2}, 1right)).

Application of the Inequality

Using the triple inequality, we can find an upper bound for the integrand on [0, alpha] as follows:

[ frac{2sin(2x)}{sin(2x^4)} geq frac{frac{2}{pi} cdot 2x}{2 cdot 2x^4} frac{1}{pi x^3} ]

Next, we evaluate the integral of the simpler function frac{1}{pi x^3} over the interval [0, alpha]:

[ int_0^{alpha} frac{1}{pi x^3} , dx lim_{t to 0^ } left( -frac{1}{2pi x^2} Big|_t^{alpha} right) infty ]

This direct comparison shows that the integral in question is divergent. By the Direct Comparison Test, if the integral of the larger function diverges, then the integral of the smaller function (in this case, the given integrand) must also diverge.

Conclusion

The integral of frac{2sin(2x)}{sin(2x^4)} from 0 to pi/4 is divergent due to its behavior near x 0. The method used to show this involves a combination of the double angle identity, the Direct Comparison Test, and a careful analysis of the integrand's behavior in a small interval around zero. This analysis provides a deeper understanding of the concept of integral divergence.