7–84. Evaluate the following integrals.
49. ∫ tan³x · sec⁹x dx

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Step 1: Recognize that the integral involves powers of tangent and secant functions. To simplify, use trigonometric identities. Recall that sec²x = 1 + tan²x, which can help in substitution later.

Step 2: Break down the integral ∫ tan³x · sec⁹x dx into manageable parts. Factor out sec²x from sec⁹x to prepare for substitution: ∫ tan³x · sec⁷x · sec²x dx.

Step 3: Use substitution. Let u = tan(x), which implies that du = sec²x dx. Replace tan(x) with u and sec²x dx with du in the integral.

Step 4: Rewrite the integral in terms of u: ∫ u³ · (1 + u²)³ du. Expand (1 + u²)³ using the binomial theorem to express it as a polynomial.

Step 5: Integrate the resulting polynomial term by term with respect to u. After integration, substitute back u = tan(x) to express the solution in terms of x.

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Integration Techniques

Integration techniques are methods used to evaluate integrals, which can include substitution, integration by parts, and trigonometric identities. For integrals involving trigonometric functions like tan and sec, recognizing patterns and applying appropriate identities can simplify the process. Mastery of these techniques is essential for solving complex integrals effectively.

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Trigonometric identities are equations that relate the angles and sides of triangles, and they are crucial for simplifying expressions involving trigonometric functions. For example, the identity sec²x = 1 + tan²x can be used to express secant in terms of tangent, which is helpful in integrals involving both functions. Understanding these identities allows for easier manipulation of integrals.

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