Evaluate the integrals in Exercises 37–44.
∫ tan³(x) sec³(x) dx

Verified step by step guidance

1. Rewrite the integral \( \int \tan^{3}(x) \sec^{3}(x) \, dx \) by expressing \( \tan^{3}(x) \) as \( \tan^{2}(x) \tan(x) \). This gives \( \int \tan^{2}(x) \tan(x) \sec^{3}(x) \, dx \).
2. Use the Pythagorean identity \( \tan^{2}(x) = \sec^{2}(x) - 1 \) to rewrite \( \tan^{2}(x) \) in the integral. So the integral becomes \( \int (\sec^{2}(x) - 1) \tan(x) \sec^{3}(x) \, dx \).
3. Split the integral into two parts: \( \int \sec^{2}(x) \tan(x) \sec^{3}(x) \, dx - \int \tan(x) \sec^{3}(x) \, dx \). Simplify the powers of \( \sec(x) \) where possible.
4. Consider a substitution to simplify the integrals. Notice that the derivative of \( \sec(x) \) is \( \sec(x) \tan(x) \), so let \( u = \sec(x) \), which implies \( du = \sec(x) \tan(x) \, dx \).
5. Rewrite the integrals in terms of \( u \) and \( du \), then integrate the resulting polynomial expressions in \( u \). After integration, substitute back \( u = \sec(x) \) to express the answer in terms of \( x \).

Key Concepts

Trigonometric Identities

Trigonometric identities like \( \sec^2(x) = 1 + \tan^2(x) \) help simplify integrals involving powers of tangent and secant. Recognizing these identities allows rewriting the integrand into a more manageable form for integration.

Integration by Substitution

Integration by substitution involves changing variables to simplify the integral. For integrals with powers of tangent and secant, substituting \( u = \tan(x) \) or \( u = \sec(x) \) often transforms the integral into a polynomial form easier to integrate.

Reduction Formulas for Powers of Tangent and Secant

Reduction formulas provide a systematic way to reduce the powers of tangent and secant in an integral step-by-step. These formulas help break down complex integrals into simpler ones, making it easier to evaluate integrals like \( \int \tan^m(x) \sec^n(x) dx \).