Evaluate the given indefinite integral by a trigonometric...

Evaluate the given indefinite integral by a trigonometric substitution where appropriate. You should be able to evaluate some of the integrals without a substitution.
$$
\int \sqrt{6 x-x^{2}} d x
$$

Key Concepts

Back-Substitution

After performing a trigonometric substitution and integrating with respect to the new variable, the result must be converted back to the original variable. Back-substitution involves using the inverse of the original substitution, thereby expressing the final antiderivative in terms of the original variable.

Trigonometric Substitution

Trigonometric substitution is a method used to evaluate integrals involving square roots of quadratic expressions, especially those in the form ?(a² - u²), ?(u² - a²), or ?(u² + a²). By substituting u with a trigonometric function (such as a sin?, a sec?, or a tan?), the integral is transformed into one involving trigonometric identities that are easier to integrate.

Indefinite Integration

Indefinite integration is the process of finding the antiderivative of a function, which results in a family of functions differing by an arbitrary constant. It involves reversing the process of differentiation and is often used to determine a general solution to a problem involving rates of change.

Algebraic Manipulation and Completing the Square

Before applying substitution techniques, it is sometimes necessary to rewrite the expression under the square root in a form that is easier to integrate. Completing the square involves rearranging a quadratic expression into a perfect square plus or minus a constant, which can reveal a standard integral form, such as the square root of a squared quantity minus a variable squared.

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