Simplifying Square Root Expressions Using Trigonometric Substitution

Using trigonometric substitution, simplify the expression $\sqrt{a^2 - u^2}$, where $u = a \sin\theta$.

Trigonometric substitution is a technique used in calculus that simplifies the integration process by exchanging variables in a given expression. This method is particularly useful for dealing with expressions involving the square root of quadratic polynomials, such as the one presented in this problem. The key idea here is exploiting the Pythagorean identities from trigonometry, which are handy tools when squaring sines and cosines, as they naturally relate back to one or zero, aligning with common algebraic constraints.

In this specific problem, the expression $\sqrt{a^2 - u^2}$ is transformed using the substitution $u = a \sin\theta$. This substitution is derived from the identity $\sin^2\theta + \cos^2\theta = 1$, allowing $a^2 - u^2$ to be rewritten in terms of a single trigonometric function. The substitution simplifies the integral potentially present, by eliminating the square root through simplification, thereby easing the process to solve or further manipulate the integral.

Understanding when and how to apply trigonometric substitution is a valuable skill, especially when faced with integrals involving radicals or equations that seem daunting initially. Recognizing these scenarios and confidently applying trigonometric identities can transform a seemingly complex calculus problem into a more manageable form.

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