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Chapter 6: Problem 1

Decide whether the integral is improper. Explain your reasoning. $$ \int_{0}^{1} \frac{d x}{3 x-2} $$

Short Answer

Expert verified
Yes, the given integral \( \int_{0}^{1} \frac{d x}{3 x-2} \) is improper because the function becomes undefined for \( x = \frac{2}{3}\), which lies within the interval of integration.

Step by step solution

01

Recognize any Instances of Undefined Function within the Interval.

To determine if the integral is improper or not, it is important to identify if the function is defined for all values within the interval of integration, which in this case is from 0 to 1. When we set the denominator of the integrand to zero, we get \( x = \frac{2}{3}\). This value lies within the interval [0, 1]. Hence, we have an undefined value within the interval of integration.
02

Identify if it is Improper Integral

A singularity in the function, where the denominator equals zero (\( x = \frac{2}{3}\)) within the interval of integration, makes this integral improper. In this case, the function becomes unbounded in the interval of integration, which is a condition for the integral to be considered improper.
03

Conclusion

Based on the analysis, it's found that the integrand becomes undefined for \( x = \frac{2}{3}\), within the interval of integration. So, the given integral \( \int_{0}^{1} \frac{d x}{3 x-2} \) is indeed improper as it has a singularity within the interval of integration.

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Most popular questions from this chapter

Present Value Use a program similar to the Simpson's Rule program on page 454 with \(n=8\) to approximate the present value 454 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 \(6.1 .)\) $$ c(t)=6000+200 \sqrt{t}, r=7 \%, t_{1}=4 $$

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

Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{-\infty}^{0} \frac{x}{x^{2}+1} d x $$

Use a spreadsheet to complete the table for the specified values of and to demonstrate that $$ \lim _{x \rightarrow \infty} x^{n} e^{-a x}=0, \quad a>0, n>0 $$ $$ \begin{array}{|c|c|c|c|c|}\hline x & {1} & {10} & {25} & {50} \\ \hline x^{n} e^{-a x} & {} & {} & {} & {} \\ \hline\end{array} $$ $$ a=\frac{1}{2}, n=2 $$

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