Chapter 12: Problem 46
Evaluate the definite integral. $$ \int_{2}^{4} \frac{x^{2}}{(3 x-5)} d x $$
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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} $$ a=1, n=1 $$
Use a program similar to the Simpson's Rule program on page 906 to approximate the integral. Use \(n=100\). $$ \int_{1}^{4} x \sqrt{x+4} d x $$
Use a program similar to the Simpson's Rule program on page 906 to approximate the integral. Use \(n=100\). $$ \int_{1}^{4} x^{2} \sqrt{x+4} d x $$
Use the error formulas to find bounds for the error in approximating the integral using (a) the Trapezoidal Rule and (b) Simpson's Rule. (Let \(n=4 .\).) $$ \int_{0}^{2} x^{3} d x $$
Decide whether the integral is improper. Explain your reasoning. $$ \int_{1}^{3} \frac{d x}{x^{2}} $$
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