91Ó°ÊÓ

Evaluate the limit (a) using techniques from Chapters 1 and 3 and (b) using L'Hôpital's Rule. $$ \lim _{x \rightarrow \infty} \frac{5 x^{2}-3 x+1}{3 x^{2}-5} $$

Use a table of integrals with forms involving the trigonometric functions to find the integral. $$ \int \frac{1}{1-\tan 5 x} d x $$

Use Wallis's Formulas to evaluate the integral. \(\int_{0}^{\pi / 2} \sin ^{2} x d x\)

Find the integral. (Note: Solve by the simplest method-not all require integration by parts.) $$ \int x \sqrt{x-1} d x $$

Evaluate the limit, using L'Hôpital's Rule if necessary. (In Exercise \(18, n\) is a positive integer.) $$ \lim _{x \rightarrow \infty} \frac{\ln x^{4}}{x^{3}} $$

(a) describe the type of indeterminate form (if any) that is obtained by direct substitution. (b) Evaluate the limit, using L'Hôpital's Rule if necessary. (c) Use a graphing utility to graph the function and verify the result in part (b). $$ \text { 9. } \lim _{x \rightarrow \infty}\left(x \sin \frac{1}{x}\right) $$

Use the method of partial fractions to verify the integration formula.\(\int \frac{1}{x(a+b x)} d x=\frac{1}{a} \ln \left|\frac{x}{a+b x}\right|+C\)

Determine all values of \(p\) for which the improper integral converges. $$ \int_{1}^{\infty} \frac{1}{x^{p}} d x $$

Slope Fields, a differential equation, a point, and a slope field are given. (a) Sketch two approximate solutions of the differential equation on the slope field, one of which passes through the given point. (b) Use integration to find the particular solution of the differential equation and use a graphing utility to graph the solution. Compare the result with the sketches in part (a). To print an enlarged copy of the graph, select the MathGraph button. $$ \begin{aligned} &\frac{d s}{d t}=\frac{t}{\sqrt{1-t^{4}}} \\ &\left(0,-\frac{1}{2}\right) \end{aligned} $$

Epidemic Model A single infected individual enters a community of \(n\) susceptible individuals. Let \(x\) be the number of newly infected individuals at time \(t\). The common epidemic model assumes that the disease spreads at a rate proportional to the product of the total number infected and the number not yet infected. So, \(d x / d t=k(x+1)(n-x)\) and you obtain \(\int \frac{1}{(x+1)(n-x)} d x=\int k d t\) Solve for \(x\) as a function of \(t\).