Trigonometric Substitution in Integration

Simplify the integral using trigonometric substitution and express the result back in terms of $x$.

Trigonometric substitution is a powerful technique in calculus used to simplify certain integrals, especially those involving roots of quadratic expressions. The fundamental idea behind this method is to use trigonometric identities and functions to transform the integral into a simpler form. This transformation often involves using identities like sine squared plus cosine squared equals one to replace complex algebraic expressions with more manageable trigonometric terms. The key benefit of trigonometric substitution is that it allows us to deal with integrals that would otherwise be very difficult to simplify using standard algebraic methods.

In this problem, the main strategy involves recognizing the form of the integrand and deciding on an appropriate trigonometric substitution. This decision often hinges on the expression under the square root. For example, for a term like square root of a squared minus x squared, a sine substitution might be fitting, whereas a cosine substitution could be more appropriate for square root of x squared minus a squared.

After making the initial substitution, the integral is transformed into one in terms of trigonometric functions, which can often be evaluated more directly. The process does not end there, however, as the final challenge is to convert the result back into the original variable of integration. This involves using the inverse trigonometric functions or some known trigonometric identities to express the final answer in terms of the initial variable, ensuring the integration process is fully completed.

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