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Chapter 12: Problem 6
Use the indicated formula from the table of integrals in this section to find the indefinite integral. $$ \int x^{2} \sqrt{x^{2}+9} d x, \text { Formula } 22 $$
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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} $$ a=\frac{1}{2}, n=5 $$
Capitalized Cost In Exercises 51 and 52, find the capitalized cost \(C\) of an asset \((a)\) for \(n=5\) years, \((b)\) for \(n=10\) years, and (c) forever. The capitalized cost is given by \(C=C_{0}+\int_{0}^{n} c(t) e^{-r t} d t\) where \(C_{0}\) is the original investment, \(t\) is the time in years, \(r\) is the annual interest rate compounded continuously, and \(c(t)\) is the annual cost of maintenance (measured in dollars). [Hint: For part (c), see Exercises \(35-38 .]\) $$ C_{0}=\$ 650,000, c(t)=25,000(1+0.08 t), r=12 \% $$
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. $$ \int_{0}^{2} \frac{1}{\sqrt[3]{x-1}} d x $$
Determine the amount of money required to set up a charitable endowment that pays the amount \(P\) each year indefinitely for the annual interest rate \(r\) compounded continuously. $$ P=\$ 5000, r=7.5 \% $$
Use the error formulas to find \(n\) such that the error in the approximation of the definite integral is less than \(0.0001\) using (a) the Trapezoidal Rule and (b) Simpson's Rule. $$ \int_{0}^{1} x^{3} d x $$
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