For Exercises 49–52, complete the square before using an appropriate trigonometric substitution.
∫ √(x² + 2x + 2) / (x² + 2x + 1) dx

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Start by completing the square for the expressions inside the integral. For the numerator's radicand \(x^2 + 2x + 2\), rewrite it as \(x^2 + 2x + 1 + 1\), which can be expressed as \((x + 1)^2 + 1\).

Similarly, complete the square for the denominator \(x^2 + 2x + 1\), which is already a perfect square and can be written as \((x + 1)^2\).

Rewrite the integral using these completed squares: \(\int \frac{\sqrt{(x + 1)^2 + 1}}{(x + 1)^2} \, dx\).

Make the substitution \(u = x + 1\), so the integral becomes \(\int \frac{\sqrt{u^2 + 1}}{u^2} \, du\).

Use a trigonometric substitution appropriate for \(\sqrt{u^2 + 1}\), such as \(u = \tan \theta\), which implies \(\sqrt{u^2 + 1} = \sec \theta\). Then express \(du\) in terms of \(d\theta\) and rewrite the integral accordingly.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Completing the Square

Completing the square is a technique used to rewrite quadratic expressions in the form (x + a)² + b. This simplifies the integrand and helps identify suitable substitutions, especially for integrals involving square roots of quadratic expressions.

Trigonometric Substitution

Trigonometric substitution replaces algebraic expressions with trigonometric functions to simplify integrals involving square roots of quadratic forms. The substitution depends on the form a² ± x² or x² ± a², converting the integral into a trigonometric integral that is easier to evaluate.

Integration of Rational Functions Involving Quadratics

Integrals with rational functions containing quadratic polynomials often require algebraic manipulation and substitution. Understanding how to simplify the denominator and numerator, especially after completing the square, is essential to apply substitution methods effectively.

For Exercises 49–52, complete the square before using an appropriate trigonometric substitution.∫ √(x² + 2x + 2) / (x² + 2x + 1) dx