To evaluate ∫ (1 + x^2)^(1/2) dx, what technique is required?

The integral ∫ (1 + x^2)^(1/2) dx involves an expression that resembles a form suitable for trigonometric substitution. This technique is often utilized when dealing with integrals that contain square roots of quadratic polynomials, particularly those that can be simplified by substituting ( x = \tan(\theta) ).

In this specific case, using the substitution ( x = \tan(\theta) ) transforms the integrand, as follows:

• The expression under the square root becomes ( \sqrt{1 + \tan^2(\theta)} ), which simplifies to ( \sec(\theta) ) based on the identity ( 1 + \tan^2(\theta) = \sec^2(\theta) ).

• Additionally, the derivative ( dx ) changes to ( \sec^2(\theta) d\theta ).

Thus, the integral transforms into a form that is much simpler to evaluate.

The other techniques mentioned, such as partial fraction decomposition, simple integration techniques, and integration by parts, are not appropriate for this integral due to the nature of the expression involved. Those methods are typically used for rational functions, straightforward polynomial expressions, or products of functions, but