Chapter 11: Q. 15 (page 765)

Write the partial fraction decomposition of each rational expression:
$\frac{1}{x (x^{2} + 1)}$ .

Short Answer

Expert verified

The partial fraction decomposition is,

$\frac{1}{x (x^{2} + 1)} = \frac{1}{x} + \frac{- x}{x^{2} + 1}$ .

Step by step solution

Step 1. Given Information.

The rational expression is,

$\frac{1}{x (x^{2} + 1)}$ .

Step 2. Decompose the partial fractions.

Decompose the rationals expressions to get,

$\frac{1}{x (x^{2} + 1)} = \frac{A}{x} + \frac{B x + C}{x^{2} + 1} . . . . . . . . (1)$

Multiplying both sides by $x (x^{2} + 1)$ ,

$1 = A (x^{2} + 1) + (B x + C) x 1 = A x^{2} + A + B x^{2} + C x 1 = (A + B) x^{2} + C x + A . . . . . . . . . (2)$

Equating the coefficients with like power, to get,

$A = 1 B = - 1 C = 0$

The partial fraction decomposition is,

$\frac{1}{x (x^{2} + 1)} = \frac{1}{x} + \frac{- 1 x + 0}{x^{2} + 1}$

$= \frac{1}{x} + \frac{- x}{x^{2} + 1}$

Step 3. Checking the solution.

Adding the rational expressions,

$\frac{1}{x} + \frac{- x}{x^{2} + 1} = \frac{1 (x^{2} + 1) + x (- x)}{x^{2} + 1}$

$= \frac{x^{2} + 1 - x^{2}}{x (x^{2} + 1)} = \frac{1}{x (x^{2} + 1)}$

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Write the partial fraction decomposition of each rational expression:
$\frac{1}{x (x^{2} + 1)}$ .

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Decompose the partial fractions.

Step 3. Checking the solution.

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Write the partial fraction decomposition of each rational expression:1x(x2+1).

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Decompose the partial fractions.

Step 3. Checking the solution.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Theoretical and Mathematical Physics

Decision Maths

Statistics

Probability and Statistics

Calculus

Study anywhere. Anytime. Across all devices.

Write the partial fraction decomposition of each rational expression:
$\frac{1}{x (x^{2} + 1)}$ .