(a) Make an appropriate u-substitution, and then use the...

(a) Make an appropriate u-substitution, and then use the Endpaper Integral Table to evaluate the integral. (b) If you have a CAS, use it to evaluate the integral (no substitution), and then confirm that the result is equivalent to that in part (a).
$$\int \cos \sqrt{x} d x$$

Key Concepts

Computer Algebra Systems (CAS) Verification

Computer Algebra Systems are software tools that can symbolically perform complex mathematical computations, including integration. Using a CAS to verify manually computed integrals serves as a check on the work, ensuring that the obtained result is correct and consistent with alternative methods.

Integration Tables

Integration tables are compiled lists of integrals that provide formulas for common integrals. These tables allow one to look up integrals that match a given standard form, thereby eliminating the need to derive the antiderivative from scratch. They are particularly useful after a substitution simplifies an integral into a recognizable form.

U-Substitution

U-substitution is a method for simplifying integrals by making a change of variable. It is particularly useful when an integral contains a composite function. By substituting a part of the integrand with a new variable, the integral can be transformed into a simpler form that is easier to evaluate.

Chain Rule

The chain rule is a differentiation technique that explains how to take the derivative of a composite function. In the context of integration, it helps in understanding and inverting the process when making a substitution. Recognizing how the derivative of the substituted variable relates to the original variable is critical when performing u-substitution.

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