9–61. Trigonometric integrals Evaluate the following integrals.
31. ∫ 20 tan⁶x dx

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Step 1: Recognize that the integral involves a power of the tangent function, tan⁶(x). To simplify, use trigonometric identities. Recall that tan²(x) = sec²(x) - 1, which can help reduce the power of tangent.

Step 2: Rewrite tan⁶(x) as (tan²(x))³. Substitute tan²(x) = sec²(x) - 1 into the expression, resulting in (sec²(x) - 1)³.

Step 3: Expand (sec²(x) - 1)³ using the binomial theorem: (sec²(x) - 1)³ = sec⁶(x) - 3sec⁴(x) + 3sec²(x) - 1.

Step 4: Break the integral into separate terms: ∫ 20 tan⁶(x) dx = 20 ∫ (sec⁶(x) - 3sec⁴(x) + 3sec²(x) - 1) dx. This allows you to evaluate each term individually.

Step 5: For each term, use standard integration techniques or formulas for secant powers. For example, ∫ sec⁶(x) dx, ∫ sec⁴(x) dx, ∫ sec²(x) dx, and ∫ 1 dx can be evaluated using known results or reduction formulas. Combine the results to complete the solution.

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Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, are fundamental in calculus, particularly in integration and differentiation. The tangent function, tan(x), is defined as the ratio of the sine and cosine functions, tan(x) = sin(x)/cos(x). Understanding the properties and identities of these functions is essential for manipulating and integrating trigonometric expressions.

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Introduction to Trigonometric Functions

Integration Techniques

Integration techniques are methods used to find the integral of a function. For trigonometric integrals, techniques such as substitution, integration by parts, and using trigonometric identities are often employed. In the case of ∫ tan⁶x dx, recognizing patterns and applying appropriate techniques can simplify the integration process.

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Integration by Parts for Definite Integrals

Power Reduction Formulas

Power reduction formulas are used to express higher powers of trigonometric functions in terms of lower powers. For example, the formula for tan²x can be expressed in terms of sec²x, which helps in simplifying integrals involving powers of tangent. Utilizing these formulas is crucial for evaluating integrals like ∫ tan⁶x dx, as they allow for easier integration by reducing the complexity of the expression.

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