Comparison Theorem and Convergence of Trigonometric Integral

Use the comparison theorem to determine whether the integral from 0 to $\pi$ of $\frac{\sin^2 x}{\sqrt{x}} \, dx$ is convergent or divergent.

In this problem, we are tasked with determining the convergence or divergence of the integral of a trigonometric function using the comparison theorem, a powerful tool in the analysis of integrals. The integral in question involves a sine squared function divided by the square root of x over the interval from zero to pi. The comparison theorem is used to ascertain the behavior of an integral by comparing it to another integral whose convergence is known.

To employ the comparison theorem effectively, one must identify a function that bounds the given function from above or below and whose integral over the specified interval is known to either converge or diverge. The challenge lies in selecting an appropriate comparison function, which often requires insights into the asymptotic behavior of the function near the endpoints of the interval. In this case, recognizing the behavior of the sine function and its relationship to x near zero is crucial.

Understanding convergence in the context of improper integrals is essential, particularly when approaching limits where the endpoint leads to undefined behavior, such as division by zero. This problem reinforces the importance of domain knowledge in trigonometric identities and integration techniques as part of broader strategies for testing convergence. The ability to leverage the comparison theorem hinges on both finding suitable bounds and understanding their convergence properties.

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