65-68. Reduction formulas Use the reduction formulas in a table of integrals to evaluate the following integrals.
67. ∫tan⁴(3y) dy

Verified step by step guidance

1. Step 1: Recognize that the integral involves a power of the tangent function, specifically tan⁴(3y). Reduction formulas are often used to simplify integrals of trigonometric functions raised to powers.
2. Step 2: Use the reduction formula for the integral of tanⁿ(x), which is typically expressed as: ∫tanⁿ(x) dx = (1/(n-1))tanⁿ⁻²(x) - ∫tanⁿ⁻²(x) dx, where n > 1. In this case, n = 4.
3. Step 3: Rewrite the integral ∫tan⁴(3y) dy using the reduction formula. Substitute n = 4 into the formula, and account for the chain rule due to the argument 3y. This introduces a factor of 1/3 outside the integral.
4. Step 4: Break the integral into smaller parts using the reduction formula. You will now have an expression involving ∫tan²(3y) dy. Recall that tan²(x) can be rewritten using the identity tan²(x) = sec²(x) - 1.
5. Step 5: Simplify further by substituting sec²(3y) - 1 for tan²(3y) and evaluate the resulting integrals step by step. Be mindful of constants and the chain rule throughout the process.

Key Concepts

Reduction Formulas

Reduction formulas are mathematical expressions that simplify the process of integrating functions by reducing the power of the function in each step. They are particularly useful for trigonometric functions, allowing the integral of a higher power to be expressed in terms of integrals of lower powers. This technique often involves recursive relationships that can simplify complex integrals into manageable forms.

Integration of Trigonometric Functions

Integrating trigonometric functions involves applying specific techniques and identities to find the antiderivative of functions like sine, cosine, and tangent. For example, the integral of tangent can be expressed in terms of logarithmic functions. Understanding the properties and relationships of these functions is crucial for effectively applying reduction formulas in integration.

Substitution Method

The substitution method is a technique used in calculus to simplify the integration process by changing variables. This method involves substituting a part of the integral with a new variable, which can make the integral easier to evaluate. In the context of trigonometric integrals, this often involves substituting a trigonometric identity or a function of the variable to facilitate the integration process.