Integrating with repeated linear factors Evaluate f (x) dr, where5x² – 3x + 2f(x)x – 2x?

f(x)=5x2-3x+2x3-2x2 To find: ∫f(x) dx [By partial fractions]

Answered: Integrating with repeated linear…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

See similar textbooks

Related questions

Question

Integrating with repeated linear factors Evaluate f (x) dr, where
5x² – 3x + 2
f(x)
x – 2x?

Expert Solution

Step 1: Given

$f (x) = \frac{5 x^{2} - 3 x + 2}{x^{3} - 2 x^{2}}$

To find:

$\int f (x) d x$ [By partial fractions]

Step 2:

A rational function f(x) is a ratio of two polynomials $\frac{p (x)}{q (x)}$ , where q(x) ≠ 0.

If the degree of p(x) is lesser than the degree of q(x), then it is a proper fraction, else it is an improper fraction.

The process of taking a rational expression (improper fraction) and decomposing it into simpler rational expressions that we can add or subtract to get the original rational expression is called partial fraction decomposition.

For each factor in the q(x) we can use the following table to determine the term(s) we pick up in the partial fraction decomposition.

Factor in denominator	Term in partial fraction decomposition

Step 3

$f (x) = \frac{5 x^{2} - 3 x + 2}{x^{3} - 2 x^{2}}$ is a improper fraction.

$f (x) = \frac{5 x^{2} - 3 x + 2}{x^{3} - 2 x^{2}} = \frac{5 x^{2} - 3 x + 2}{x^{2} (x - 2)}$

We can decompose it as

$\begin{array}{rcl} \frac{5 x^{2} - 3 x + 2}{x^{2} (x - 2)} & = & \frac{A}{x} + \frac{B}{x^{2}} + \frac{C}{(x - 2)} \\ \frac{5 x^{2} - 3 x + 2}{x^{2} (x - 2)} & = & [\frac{A}{x} \times \frac{x (x - 2)}{x (x - 2)}] + [\frac{B}{x^{2}} \times \frac{(x - 2)}{(x - 2)}] + [\frac{C}{(x - 2)} \times \frac{x^{2}}{x^{2}}] \\ = & [\frac{A x (x - 2)}{x^{2} (x - 2)}] + [\frac{B (x - 2)}{x^{2} (x - 2)}] + [\frac{C x^{2}}{x^{2} (x - 2)}] \\ \frac{5 x^{2} - 3 x + 2}{x^{2} (x - 2)} & = & \frac{A x (x - 2) + B (x - 2) + C x^{2}}{x^{2} (x - 2)} \\ 5 x^{2} - 3 x + 2 & = & A x (x - 2) + B (x - 2) + C x^{2} . . . (1) \end{array}$