In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (8x² + 8x + 2) / (4x² + 1)² dx

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Recognize that the integrand is a rational function where the denominator is a repeated quadratic factor: \((4x^{2} + 1)^{2}\). Since the denominator is a repeated irreducible quadratic, set up the partial fraction decomposition accordingly.

Express the integrand as a sum of partial fractions of the form: \(\frac{8x^{2} + 8x + 2}{(4x^{2} + 1)^{2}} = \frac{Ax + B}{4x^{2} + 1} + \frac{Cx + D}{(4x^{2} + 1)^{2}}\), where \(A\), \(B\), \(C\), and \(D\) are constants to be determined.

Multiply both sides of the equation by \((4x^{2} + 1)^{2}\) to clear the denominators, resulting in: \(8x^{2} + 8x + 2 = (Ax + B)(4x^{2} + 1) + (Cx + D)\).

Expand the right-hand side and collect like terms in powers of \(x\). Then, equate the coefficients of corresponding powers of \(x\) 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 \(D\). Once found, rewrite the integral as the sum of simpler integrals involving these partial fractions, which can then be integrated using standard techniques such as substitution and recognizing derivatives of inverse trigonometric functions.

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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 express a rational function as a sum of simpler fractions. This method is especially useful when integrating rational functions, as it breaks down complex expressions into manageable parts that can be integrated individually.

Integration of Rational Functions

Integrating rational functions often involves rewriting the integrand into simpler terms, such as polynomials or partial fractions. Recognizing when to use substitution, partial fractions, or other integration techniques is essential for solving these integrals effectively.

Handling Repeated Quadratic Factors

When the denominator contains repeated quadratic factors, partial fractions include terms with increasing powers of the quadratic in the denominator. Understanding how to set up and solve for coefficients in these cases is crucial for correctly decomposing and integrating the function.

In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.∫ (8x² + 8x + 2) / (4x² + 1)² dx