7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
54. ∫ y⁴/(1 + y²) dy

Verified step by step guidance

1. Step 1: Recognize that the integral ∫ y⁴/(1 + y²) dy involves a rational function with a quadratic term in the denominator. This suggests that trigonometric substitution might simplify the integral. Specifically, use the substitution y = tan(θ), which implies dy = sec²(θ) dθ and 1 + y² = sec²(θ).
2. Step 2: Substitute y = tan(θ) into the integral. Replace y⁴ with (tan(θ))⁴, 1 + y² with sec²(θ), and dy with sec²(θ) dθ. The integral becomes ∫ (tan⁴(θ) / sec²(θ)) sec²(θ) dθ, which simplifies to ∫ tan⁴(θ) dθ.
3. Step 3: Simplify the integral further by expressing tan⁴(θ) in terms of lower powers of tan(θ). Use the identity tan²(θ) = sec²(θ) - 1 to rewrite tan⁴(θ) as (tan²(θ))² = (sec²(θ) - 1)². Expand this expression to get tan⁴(θ) = sec⁴(θ) - 2sec²(θ) + 1.
4. Step 4: Break the integral ∫ tan⁴(θ) dθ into separate integrals using the expanded form: ∫ tan⁴(θ) dθ = ∫ (sec⁴(θ) - 2sec²(θ) + 1) dθ. Evaluate each term separately: ∫ sec⁴(θ) dθ, ∫ sec²(θ) dθ, and ∫ 1 dθ. Recall that ∫ sec²(θ) dθ = tan(θ) and ∫ 1 dθ = θ. For ∫ sec⁴(θ) dθ, use reduction formulas or known results.
5. Step 5: After evaluating the integral in terms of θ, revert back to the original variable y using the substitution y = tan(θ). Use the relationship tan(θ) = y and θ = arctan(y) to express the final result in terms of y.

Key Concepts

Trigonometric Substitution

Trigonometric substitution is a technique used in calculus to simplify integrals involving algebraic expressions. By substituting a variable with a trigonometric function, such as y = tan(θ) or y = sin(θ), the integral can often be transformed into a more manageable form. This method is particularly useful for integrals that include square roots or rational functions, allowing for easier integration.

Integral of Rational Functions

Integrating rational functions involves finding the antiderivative of a function that is the ratio of two polynomials. In the case of the integral ∫ y⁴/(1 + y²) dy, recognizing the structure of the rational function is crucial. Techniques such as polynomial long division or partial fraction decomposition may be employed, but trigonometric substitution can also simplify the process by transforming the integral into a trigonometric form.

Differentiation of Trigonometric Functions

Understanding the derivatives of trigonometric functions is essential when performing trigonometric substitution. For example, if y = tan(θ), then dy = sec²(θ) dθ. This relationship is vital for converting the integral back into terms of θ after substitution. Familiarity with these derivatives ensures that the integration process remains accurate and efficient, allowing for the correct evaluation of the integral.