7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.
16. ∫ x²/(25 + x²)² dx

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Step 1: Recognize that the integral involves a term of the form (a² + x²). This suggests using the trigonometric substitution x = a * tan(θ), where a = 5 in this case. Substitute x = 5 * tan(θ), which implies dx = 5 * sec²(θ) dθ.

Step 2: Substitute x = 5 * tan(θ) into the integral. Replace x² with (5 * tan(θ))² = 25 * tan²(θ), and replace dx with 5 * sec²(θ) dθ. The denominator (25 + x²)² becomes (25 + 25 * tan²(θ))² = (25 * sec²(θ))².

Step 3: Simplify the integral using the trigonometric identities. The integral becomes ∫ (25 * tan²(θ)) / (625 * sec⁴(θ)) * (5 * sec²(θ)) dθ. Simplify the expression by canceling terms and reducing powers of sec(θ).

Step 4: After simplification, the integral reduces to ∫ (tan²(θ) / sec²(θ)) dθ. Use the identity tan²(θ) = sec²(θ) - 1 to rewrite the integral as ∫ (sec²(θ) - 1) / sec²(θ) dθ = ∫ (1 - 1/sec²(θ)) dθ.

Step 5: Split the integral into two parts: ∫ 1 dθ - ∫ 1/sec²(θ) dθ. Evaluate these integrals separately. The first integral ∫ 1 dθ is θ, and the second integral ∫ 1/sec²(θ) dθ is tan(θ). Finally, back-substitute θ using the original substitution x = 5 * tan(θ) to express the result in terms of x.

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Trigonometric Substitution

Trigonometric substitution is a technique used in calculus to simplify integrals involving square roots or quadratic expressions. By substituting a variable with a trigonometric function, such as x = a tan(θ) or x = a sin(θ), the integral can often be transformed into a more manageable form. This method leverages the identities of trigonometric functions to facilitate integration.

Integral of Rational Functions

Integrating rational functions involves finding the antiderivative of a fraction where both the numerator and denominator are polynomials. Techniques such as polynomial long division, partial fraction decomposition, or trigonometric substitution can be employed to simplify the integral. Understanding how to manipulate these functions is crucial for solving integrals like ∫ x²/(25 + x²)² dx.

Pythagorean Identity

The Pythagorean identity is a fundamental relationship in trigonometry that states sin²(θ) + cos²(θ) = 1. This identity is often used in trigonometric substitution to relate different trigonometric functions and simplify expressions. When substituting variables, recognizing how to apply this identity can help in transforming the integral into a solvable form.

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.16. ∫ x²/(25 + x²)² dx