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Chapter 12: Problem 4

Write the partial fraction decomposition for the expression. $$ \frac{10 x+3}{x^{2}+x} $$

Short Answer

Expert verified
The partial fraction decomposition of the given expression is \( \frac{3}{x} + \frac{7}{x + 1} \).

Step by step solution

01

Break down the expression into partial fractions

When dealing with a fraction whose denominator is a polynomial that can be factored, the first step is to write the fraction as a sum of simpler fractions. For the given exercise, the denominator \(x^{2} + x\) can be factored as \(x(x + 1)\). Therefore, we write the expression as follows: \[\frac{10x + 3}{x(x + 1)} = \frac{A}{x} + \frac{B}{x + 1}\] for some constants A and B.
02

Obtain common denominators and simplify

We multiply both sides of the equation by the common denominator \(x(x + 1)\) to get rid of the fractions. Doing this gives us: \[10x + 3 = A(x + 1) + Bx\]
03

Set up system of equations to find constants A and B

We can find the values of A and B by equating coefficients of like powers of x on both sides of the equation. This gives us: \(10 = B + A\) for the coefficient of x and \(3 = A\) for the constant term. Therefore, we have two equations, A = 3 and B + 3 = 10.
04

Solve for constants A and B

Solving for A and B gives us A = 3 and B = 10 - 3 = 7. Therefore, the decomposition of the original fraction is: \[\frac{10x + 3}{x(x + 1)} = \frac{3}{x} + \frac{7}{x + 1}\]

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Most popular questions from this chapter

Approximate the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule for the indicated value of \(n\). (Round your answers to three significant digits.) $$ \int_{0}^{2} \sqrt{1+x^{3}} d x, n=4 $$

Approximate the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule for the indicated value of \(n\). (Round your answers to three significant digits.) $$ \int_{0}^{2} e^{-x^{2}} d x, n=4 $$

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges, and check your results with the results obtained by using the integration capabilities of a graphing utility. $$ \int_{0}^{1} \frac{1}{x} d x $$

Approximate the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule for the indicated value of \(n\). (Round your answers to three significant digits.) $$ \int_{0}^{1} \sqrt{1-x^{2}} d x, n=8 $$

Explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converses. $$ \int_{0}^{\infty} e^{-x} d x $$
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