23-64. Integration Evaluate the following integrals.
60.∫ 1/[(y² + 1)(y² + 2)] dy

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Recognize that the integral involves a rational function with quadratic factors in the denominator: \(\int \frac{1}{(y^{2} + 1)(y^{2} + 2)} \, dy\).

Use the method of partial fraction decomposition to express the integrand as a sum of simpler fractions. Assume a form: \(\frac{1}{(y^{2} + 1)(y^{2} + 2)} = \frac{A y + B}{y^{2} + 1} + \frac{C y + D}{y^{2} + 2}\), where \(A\), \(B\), \(C\), and \(D\) are constants to be determined.

Multiply both sides of the equation by \((y^{2} + 1)(y^{2} + 2)\) to clear the denominators, resulting in an identity involving polynomials: \(1 = (A y + B)(y^{2} + 2) + (C y + D)(y^{2} + 1)\).

Expand the right-hand side and collect like terms by powers of \(y\). Equate the coefficients of corresponding powers of \(y\) on both sides to form a system of equations for \(A\), \(B\), \(C\), and \(D\).

Solve the system of equations to find the values of \(A\), \(B\), \(C\), and \(D\). Then rewrite the integral as the sum of two simpler integrals, each involving terms like \(\int \frac{y}{y^{2} + a} \, dy\) or \(\int \frac{1}{y^{2} + a} \, dy\), which can be integrated using standard formulas.

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주요 개념

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

Partial Fraction Decomposition

Partial fraction decomposition is a technique used to break down complex rational functions into simpler fractions that are easier to integrate. For integrals involving products of quadratic terms, expressing the integrand as a sum of simpler rational expressions allows straightforward integration.

Integration of Rational Functions with Quadratic Denominators

Integrating rational functions with quadratic denominators often involves recognizing standard integral forms, such as arctangent functions. When denominators are irreducible quadratics, the integral typically results in logarithmic or inverse trigonometric functions.

Substitution Method in Integration

The substitution method simplifies integrals by changing variables to transform the integral into a more familiar form. In this problem, after partial fraction decomposition, substitution may be used to integrate terms involving expressions like 1/(y² + a²).

23-64. Integration Evaluate the following integrals.60.∫ 1/[(y² + 1)(y² + 2)] dy