Trigonometric Substitution with Square Root Function

Perform the trigonometric substitution for the integral involving $\sqrt{1 + x^2}$.

Trigonometric substitution is a technique often used in integral calculus to simplify integrals involving expressions like the square root of a sum or difference of squares. This method involves substituting a trigonometric function for a variable to exploit the Pythagorean identities, which can simplify the integrand into a form that's more readily integrable. The choice of substitution depends on the form of the expression under the square root. Common substitutions involve setting variables proportional to sine, cosine, or tangent functions, corresponding to the identities: one minus sine squared equals cosine squared, one plus tangent squared equals secant squared, or one plus sine squared equals hyperbolic cosine squared, respectively. In this problem, the expression beneath the square root is of the form one plus x squared, suggesting a hyperbolic substitution or the use of tangent due to the identity one plus tangent squared equals secant squared. Tangent or hyperbolic tangent substitutions simplify the algebraic structure to a straightforward form, typically involving linear terms instead of quadratic. Transforming the integral through such substitution often involves computing dx in terms of the trigonometric function’s derivative and simplifying using trigonometric identities. Understanding the conceptual underpinning of trigonometric substitution aids in selecting the most effective substitution and applying the technique accurately. It is essential to recognize when to reverse the substitution to return any variables to their original context in the course of evaluating a definite integral, ensuring the bounds are appropriately adjusted.

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