91Ó°ÊÓ

In Exercises 19–28, use the properties of logarithms to expand the logarithmic expression. $$ \ln \left(x \sqrt{x^{2}+5}\right) $$

Use the derivative to determine whether the function is strictly monotonic on its entire domain and therefore has an inverse function. \(f(x)=2-x-x^{3}\)

Solving an Equation In Exercises 19-24, solve for \(x\) or \(b\) . $$ \begin{array}{l}{\text { (a) } \log _{3} x+\log _{3}(x-2)=1} \\ {\text { (b) } \log _{10}(x+3)-\log _{10} x=1}\end{array} $$

Evaluating a Definite Integral In Exercises \(21-32\) evaluate the definite integral. $$ \int_{\sqrt{3}}^{3} \frac{1}{x \sqrt{4 x^{2}-9}} d x $$

Asymptotes Use a graphing utility to graph the function. Use the graph to determine any asymptotes of the function. (a) \(f(x)=\frac{8}{1+e^{-0.5 x}} \quad\) (b) \(g(x)=\frac{8}{1+e^{-0.5 / x}}\)

Use the derivative to determine whether the function is strictly monotonic on its entire domain and therefore has an inverse function. \(f(x)=x^{3}-6 x^{2}+12 x\)

Finding an Indefinite Integral In Exercises \(1-26,\) find the indefinite integral.. $$ \int \frac{1}{x^{2 / 3}\left(1+x^{1 / 3}\right)} d x $$

In Exercises 19–28, use the properties of logarithms to expand the logarithmic expression. $$ \ln \sqrt{a-1} $$

In Exercises 23–32, find the derivative of the function. $$ f(x)=\cosh (8 x+1) $$

Evaluate each expression without using a calculator. (Hint: See Example 3.) (a) \(\sec \left[\arctan \left(-\frac{3}{5}\right)\right]\) (b) \(\tan \left[\arcsin \left(-\frac{5}{6}\right)\right]\)