Trigonometric Substitution with Inverse Substitution

Perform a trigonometric substitution for evaluating the integral involving inverse substitution where $x=\frac{1}{2}u$.

Trigonometric substitution is a powerful technique used to evaluate integrals that involve expressions like the square root of a quadratic in terms of x. It involves substituting a trigonometric function for a variable, which can simplify the integral into a more solvable form. In this problem, you are tasked with using trigonometric substitution in conjunction with inverse substitution, where x is replaced by $\frac{1}{2}$ of another variable u. This introduces an additional layer of substitution that can make the integral more tractable but requires careful attention to the changes in differential as well as limits of integration if they are involved.

When performing trigonometric substitution, it's essential to recognize the form of the expression within the integral and determine the appropriate trigonometric identity that can simplify it. Common substitutions involve using sin, cos, or tangent functions to replace expressions involving square roots. The goal is to transform the integral into a form that makes use of the simpler trigonometric identities, turning it into a more standard form that is easier to integrate.

Inverse substitution, as in this problem where x is replaced by $\frac{1}{2}u$, requires expressing the original variable x in terms of a new variable u, which might further simplify the integration process. These combined techniques highlight the importance of strategic thinking in calculus, as selecting the correct form of substitution can significantly impact the ease with which you solve integrals. Keep in mind the necessary adjustments to the differential dx when changing variables, as these adjustments are crucial for correctly evaluating the integral.

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