Integral of Square Root of 9 minus x Squared using Sine Substitution

Compute the integral of $\displaystyle \int \sqrt{9 - x^2} \, dx$ using the sine substitution where $x = 3\sin(\theta)$.

In this problem, you are asked to perform integration using a technique known as trigonometric substitution. This method is particularly useful when dealing with integrals involving the square root of expressions like $a^2 - x^2$. The key idea is to convert the variable $x$ into a trigonometric function, which simplifies the integral into a more recognizable form. In this case, the substitution $x = 3\sin(\theta)$ is used. This substitution exploits the identity $\sin^2(\theta) + \cos^2(\theta) = 1$, which is fundamental in trigonometry.

Upon substituting, the integral transforms into an expression involving $\theta$. The integral can then be handled using basic trigonometric integrals, which are usually easier to evaluate. It is important to remember to change the limits of integration if the original integral was a definite one, as these limits now pertain to theta, not $x$.

Finally, converting back from $\theta$ to $x$ involves using the inverse trigonometric function and is an essential step to revert to the original variable. This method not only enhances your understanding of trigonometric functions within calculus but also enriches your toolkit for tackling complex integrals.

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