Evaluate the integrals in Exercises 33–52.
∫ sec⁶(x) dx

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Recognize that the integral involves a high even power of secant: \(\int \sec^{6}(x) \, dx\). To simplify, express \(\sec^{6}(x)\) as \(\sec^{4}(x) \cdot \sec^{2}(x)\).

Rewrite \(\sec^{4}(x)\) as \((\sec^{2}(x))^{2}\), so the integral becomes \(\int (\sec^{2}(x))^{2} \cdot \sec^{2}(x) \, dx = \int \sec^{4}(x) \cdot \sec^{2}(x) \, dx\).

Use the identity \(\sec^{2}(x) = 1 + \tan^{2}(x)\) to express powers of secant in terms of tangent, or alternatively, use the reduction formula for powers of secant: \(\int \sec^{n}(x) \, dx = \frac{\sec^{n-2}(x) \tan(x)}{n-1} + \frac{n-2}{n-1} \int \sec^{n-2}(x) \, dx\) for \(n > 1\).

Apply the reduction formula with \(n=6\) to reduce the integral \(\int \sec^{6}(x) \, dx\) to an expression involving \(\int \sec^{4}(x) \, dx\).

Repeat the reduction process on \(\int \sec^{4}(x) \, dx\) until you reach integrals of \(\sec^{2}(x)\), which is straightforward to integrate, and then combine all parts to express the original integral.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Integration of Powers of Secant

Integrating powers of secant functions often requires using reduction formulas or expressing secant in terms of tangent and secant to simplify the integral. For even powers, it is common to separate one sec²(x) factor and use trigonometric identities to reduce the power step-by-step.

Trigonometric Identities

Key identities such as sec²(x) = 1 + tan²(x) help transform the integral into a more manageable form. These identities allow substitution and reduction of powers, making it easier to integrate complex trigonometric expressions.

Substitution Method

Substitution is often used when the integral contains a function and its derivative, such as tan(x) and sec²(x). By substituting u = tan(x), the integral can be rewritten in terms of u, simplifying the integration process.

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