Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [(3v − 7) / ((v − 1)(v − 2)(v − 3))] dv

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Identify the integral to be solved: \(\int \frac{3v - 7}{(v - 1)(v - 2)(v - 3)} \, dv\).

Recognize that the integrand is a rational function where the degree of the numerator is less than the degree of the denominator, so partial fraction decomposition is appropriate.

Set up the partial fraction decomposition: express \(\frac{3v - 7}{(v - 1)(v - 2)(v - 3)}\) as \(\frac{A}{v - 1} + \frac{B}{v - 2} + \frac{C}{v - 3}\), where \(A\), \(B\), and \(C\) are constants to be determined.

Multiply both sides of the equation by the denominator \((v - 1)(v - 2)(v - 3)\) to clear the fractions, resulting in an equation involving \(3v - 7\) and the linear combination of \(A(v - 2)(v - 3)\), \(B(v - 1)(v - 3)\), and \(C(v - 1)(v - 2)\).

Solve for the constants \(A\), \(B\), and \(C\) by substituting convenient values for \(v\) (such as \(v=1\), \(v=2\), and \(v=3\)) or by equating coefficients, then integrate each resulting simpler fraction separately.

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주요 개념

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Partial Fraction Decomposition

Partial fraction decomposition is a technique used to break down a complex rational function into simpler fractions that are easier to integrate. It involves expressing the integrand as a sum of fractions with linear or quadratic denominators, which can then be integrated individually.

Integration of Rational Functions

Integrating rational functions often requires rewriting the integrand into a simpler form, such as partial fractions. Once decomposed, each term corresponds to a standard integral form, allowing the use of basic integration rules like logarithmic or arctangent integrals.

Substitution Method

The substitution method simplifies integrals by changing variables to transform the integral into a more manageable form. It is especially useful when the integrand contains composite functions or when the derivative of a function inside the integral appears elsewhere in the integrand.

Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.∫ [(3v − 7) / ((v − 1)(v − 2)(v − 3))] dv