Evaluate the integrals in Exercises 33–52.
∫ sec(x) tan²(x) dx

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Recall the trigonometric identity: \(\tan^2(x) = \sec^2(x) - 1\). This allows us to rewrite the integral in terms of secant functions.

Rewrite the integral as \(\int \sec(x) \tan^2(x) \, dx = \int \sec(x) (\sec^2(x) - 1) \, dx\).

Distribute \(\sec(x)\) inside the integral to get \(\int (\sec^3(x) - \sec(x)) \, dx\).

Split the integral into two separate integrals: \(\int \sec^3(x) \, dx - \int \sec(x) \, dx\).

Use known integration techniques: recall that \(\int \sec(x) \, dx\) is a standard integral, and for \(\int \sec^3(x) \, dx\), use integration by parts or a reduction formula to express it in terms of \(\int \sec(x) \, dx\).

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주요 개념

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Trigonometric Identities

Trigonometric identities like tan²(x) = sec²(x) - 1 help simplify integrals involving trigonometric functions. Recognizing and applying these identities can transform complex expressions into more manageable forms for integration.

Integration Techniques for Trigonometric Functions

Integrating functions involving sec(x) and tan(x) often requires substitution or rewriting the integrand using identities. Familiarity with standard integrals such as ∫ sec(x) dx and ∫ sec(x) tan(x) dx is essential for solving these problems efficiently.

Substitution Method

The substitution method involves choosing a part of the integrand as a new variable to simplify the integral. For example, setting u = sec(x) or u = tan(x) can reduce the integral to a basic form, making it easier to evaluate.

Evaluate the integrals in Exercises 33–52.∫ sec(x) tan²(x) dx