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Chapter 12: Problem 24

Use partial fractions to find the indefinite integral. $$ \int \frac{x+1}{x^{2}+4 x+3} d x $$

Short Answer

Expert verified
The solution to the integral is \(\ln|x+3| + C\).

Step by step solution

01

Factoring the denominator

First, factorize the denominator into two linear factors: \(x^{2}+4x+3\) can be factored into \((x+1)(x+3)\)
02

Writing the fraction as a sum of partial fractions

Rewrite \(\frac{x+1}{x^{2}+4 x+3}\) as \(\frac{A}{x+1}+\frac{B}{x+3}\) for some constants A and B. Multiplying through by the common denominator gives \(x+1=A(x+3)+B(x+1)\)
03

Solving for A and B

To determine A and B, you can set up a system of equations by equating the coefficients on each side of the equation. This yields the system: A+B=1, 3A+B=1. Solving the system gives A=0, B=1.
04

Integrating

Replace A and B in the partial fraction decomposition and integrate term by term to obtain: \(\int \frac{0}{x+1} dx + \int \frac{1}{x+3} dx = 0 + \ln|x+3| + C\) where C is the constant of integration.

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Most popular questions from this chapter

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. $$ \int_{0}^{2} \frac{1}{(x-1)^{4 / 3}} d x $$

Decide whether the integral is improper. Explain your reasoning. $$ \int_{1}^{\infty} x^{2} d x $$

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.) $$ \int_{0}^{3} \frac{x}{2+x+x^{2}} d x, n=6 $$

Present Value In Exercises 25 and 26, use a program similar to the Simpson's Rule program on page 906 with \(n=8\) to approximate the present value of the income \(c(t)\) over \(t_{1}\) years at the given annual interest rate \(r\). Then use the integration capabilities of a graphing utility to approximate the present value. Compare the results. (Present value is defined in Section 12.1.) $$ c(t)=6000+200 \sqrt{t}, r=7 \%, t_{1}=4 $$

Explain why the integral is improper and determine whether it diverges or converges. Evaluate the integral if it converses. $$ \int_{0}^{2} \frac{1}{(x-1)^{2 / 3}} d x $$
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