Compute the following integrals using the guidelines for...

Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may be done using techniques of integration learned previously.)
$$
\int \cos ^{5} x d x
$$

Key Concepts

Integrals of Powers of Trigonometric Functions

This concept involves the strategies used to evaluate integrals where the integrand is a trigonometric function raised to a power. The general approach depends on whether the power is even or odd. For even powers, power-reduction formulas or half-angle identities are typically used, whereas for odd powers, one often reserves one factor to facilitate substitution after using identities to convert the remaining even powers.

Substitution Method in Trigonometric Integrals

The substitution method, often referred to as u-substitution, is crucial when integrating functions that are the product of a function and its derivative. In trigonometric integrals, if one part of the integrand is the derivative of another part (such as when an odd power allows one cosine or sine factor to be separated from the rest), the substitution method can simplify the integration process by reducing the integrand to a polynomial form.

Trigonometric Identities for Simplification

Trigonometric identities, such as the Pythagorean identities and power reduction identities, are foundational tools that enable the rewriting of integrals into forms that are easier to integrate. These identities allow us to convert expressions involving high powers of sine or cosine into equivalent expressions that have lower powers or are expressed in terms of another trigonometric function, facilitating further techniques like substitution or standard integration methods.

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