Chapter 12: Problem 1
Decide whether the integral is improper. Explain your reasoning. $$ \int_{0}^{1} \frac{d x}{3 x-2} $$
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Use a spreadsheet to complete the table for the specified values of \(a\) and \(n\) to demonstrate that \(\lim _{x \rightarrow \infty} x^{n} e^{-a x}=0, \quad a>0, n>0\) \begin{tabular}{|l|l|l|l|l|} \hline\(x\) & 1 & 10 & 25 & 50 \\ \hline\(x^{n} e^{-a x}\) & & & & \\ \hline \end{tabular} $$ a=\frac{1}{2}, n=5 $$
Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{-\infty}^{\infty} 2 x e^{-3 x^{2}} d x $$
Medicine A body assimilates a 12 -hour cold tablet at a rate \(\quad\) modeled \(\quad\) by \(\quad d C / d t=8-\ln \left(t^{2}-2 t+4\right)\), \(0 \leq t \leq 12\), where \(d C / d t\) is measured in milligrams per hour and \(t\) is the time in hours. Use Simpson's Rule with \(n=8\) to estimate the total amount of the drug absorbed into the body during the 12 hours.
Determine whether the improper integral diverges or converges. Evaluate the integral if it converges. $$ \int_{1}^{\infty} \frac{1}{x^{2}} d x $$
Decide whether the integral is improper. Explain your reasoning. $$ \int_{1}^{\infty} x^{2} d x $$
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