Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [(x + 1) / (x² (x − 1))] dx

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Start by examining the integrand \( \frac{x + 1}{x^{2}(x - 1)} \). Notice that the denominator is a product of powers of \( x \) and \( (x - 1) \), suggesting that partial fraction decomposition could be a useful method.

Set up the partial fraction decomposition for \( \frac{x + 1}{x^{2}(x - 1)} \) as follows: \(\n\[\n\)\( \frac{x + 1}{x^{2}(x - 1)} = \frac{A}{x} + \frac{B}{x^{2}} + \frac{C}{x - 1} \), \(\n\]\nwhere\) \( A \), \( B \), and \( C \) are constants to be determined.

Multiply both sides of the equation by the common denominator \( x^{2}(x - 1) \) to clear the fractions: \(\n\)\(\n\)\( x + 1 = A x (x - 1) + B (x - 1) + C x^{2} \).

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 \( A \), \( B \), and \( C \).

Solve the system of equations to find the values of \( A \), \( B \), and \( C \). Once found, rewrite the integral as the sum of simpler integrals: \(\n\)\(\n\)\( \int \frac{A}{x} dx + \int \frac{B}{x^{2}} dx + \int \frac{C}{x - 1} dx \). Each of these can be integrated using basic integral formulas.

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주요 개념

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

Integration by Substitution

Integration by substitution is a method used to simplify integrals by changing variables. It involves identifying a part of the integrand as a new variable, which transforms the integral into a simpler form. This technique is especially useful when the integral contains a composite function or a complicated expression.

Partial Fraction Decomposition

Partial fraction decomposition breaks down a complex rational function into simpler fractions that are easier to integrate. This method is essential when integrating rational functions where the degree of the numerator is less than the degree of the denominator. It allows the integral to be expressed as a sum of simpler terms.

Properties of Rational Functions

Understanding rational functions, which are ratios of polynomials, is crucial for integration. Recognizing factors in the denominator and numerator helps in choosing the right method, such as substitution or partial fractions. Analyzing the function's structure guides the simplification and integration process.

Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.∫ [(x + 1) / (x² (x − 1))] dx