23-64. Integration Evaluate the following integrals.
23. ∫ [3 / ((x - 1)(x + 2))] dx

Guida verificata passo dopo passo

Start by expressing the integrand \( \frac{3}{(x - 1)(x + 2)} \) as a sum of partial fractions. Assume it can be written as \( \frac{A}{x - 1} + \frac{B}{x + 2} \), where \(A\) and \(B\) are constants to be determined.

Multiply both sides of the equation by the denominator \( (x - 1)(x + 2) \) to clear the fractions, resulting in \( 3 = A(x + 2) + B(x - 1) \).

Expand the right-hand side to get \( 3 = A x + 2A + B x - B \), then group like terms: \( 3 = (A + B) x + (2A - B) \).

Set up a system of equations by equating the coefficients of corresponding powers of \(x\) on both sides. Since the left side has no \(x\) term, the coefficient of \(x\) must be zero, and the constant term must be 3. So, \( A + B = 0 \) and \( 2A - B = 3 \).

Solve the system for \(A\) and \(B\), then rewrite the integral as \( \int \left( \frac{A}{x - 1} + \frac{B}{x + 2} \right) dx \). Finally, integrate each term separately using the formula \( \int \frac{1}{x - c} dx = \ln|x - c| + C \).

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Durata del video:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

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, allowing straightforward integration of each term.

Video consigliato:

Percorso guidato

10:07

Partial Fraction Decomposition: Distinct Linear Factors

Integration of Rational Functions

Integrating rational functions often requires rewriting the integrand into simpler parts, such as partial fractions. Once decomposed, each term can be integrated using basic integral formulas, typically involving logarithmic functions for linear denominators.

Video consigliato:

6:04

Intro to Rational Functions

Logarithmic Integration

When integrating functions of the form 1/(ax + b), the result is a logarithmic function ln|ax + b|/a plus a constant. This concept is essential for integrating the terms obtained after partial fraction decomposition with linear denominators.

Video consigliato: