Chapter 5

Find each indefinite integral. \(\int(1+10 w) \sqrt{w} d w\)

Find the Gini index for the given Lorenz curve. $$ L(x)=\frac{x+x^{2}+x^{3}}{3} $$

Find each indefinite integral by the substitution method or state that it cannot be found by our substitution formulas. $$ \int\left(3 y^{2}-6 y\right)^{3}(y-1) d y $$

Find each indefinite integral. \(\int \frac{e^{w}+w^{2}}{3} d w\)

Find the Gini index for the given Lorenz curve. $$ L(x)=0.62 x^{7.15}+0.38 x^{9.47} $$

After \(x\) practice sessions, a person can accomplish a task in \(f(x)=12 x^{-1 / 2}\) minutes. Find the average time required from the end of session 1 to the end of session 9 .

Find each indefinite integral. \(\int(1-7 w) \sqrt[3]{w} d w\)

Use a definite integral to find the area under each curve between the given \(x\) -values. For Exercises \(19-24\) also make a sketch of the curve showing the region. $$ f(x)=\frac{1}{x^{2}} \text { from } x=1 \text { to } x=3 $$

Find each indefinite integral by the substitution method or state that it cannot be found by our substitution formulas. $$ \int e^{x^{2}+2 x+5}(x+1) d x $$

Find each indefinite integral. \(\int \frac{z^{3}+z}{z^{2}} d z\)

The amount of pollution in a lake \(x\) years after the closing of a chemical plant is \(P(x)=100 / x\) tons (for \(x \geq 1\) ). Find the average amount of pollution between 1 and 10 years after the closing.

Find each indefinite integral. \(\int \frac{6 x^{3}-6 x^{2}+x}{x} d x\)

Use a definite integral to find the area under each curve between the given \(x\) -values. For Exercises \(19-24\) also make a sketch of the curve showing the region. $$ f(x)=8-4 \sqrt[3]{x} \text { from } x=0 \text { to } x=8 $$

31-32. The following tables give the distribution of family income in the United States: Exercise 31 is for the year 1977 and Exercise 32 is for \(1989 .\) Use the procedure described in the Graphing Calculator Exploration on the previous page to find the Lorenz function of the form \(x^{n}\) for the data. Then find the Gini index. If you do both problems, did family income become more concentrated or less concentrated from 1977 to \(1989 ?\) $$ \begin{array}{cc} \hline \begin{array}{c} \text { Proportion } \\ \text { (Lowest) } \\ \text { of Families } \end{array} & \begin{array}{l} \text { Proportion of } \\ \text { Income (1977) } \end{array} \\ 0.20 & 0.06 \\ 0.40 & 0.18 \\ 0.60 & 0.34 \\ 0.80 & 0.57 \\ \hline \end{array} $$

Find each indefinite integral by the substitution method or state that it cannot be found by our substitution formulas. $$ \int e^{x^{3}-3 x+7}\left(x^{2}-1\right) d x $$

Find each indefinite integral. \(\int \frac{z^{2}+1}{z} d z\)

The following tables give the distribution of family income in the United States: Exercise 31 is for the year 1977 and Exercise 32 is for \(1989 .\) Use the procedure described in the Graphing Calculator Exploration on the previous page to find the Lorenz function of the form \(x^{n}\) for the data. Then find the Gini index. If you do both problems, did family income become more concentrated or less concentrated from 1977 to \(1989 ?\) $$ \begin{array}{cc} \hline \begin{array}{c} \text { Proportion } \\ \text { (Lowest) } \\ \text { of Families } \end{array} & \begin{array}{c} \text { Proportion of } \\ \text { Income (1989) } \end{array} \\ 0.20 & 0.04 \\ 0.40 & 0.14 \\ 0.60 & 0.29 \\ 0.80 & 0.51 \\ \hline \end{array} $$

The population of the United States is predicted to be \(P(t)=310 e^{0.0073 t}\) million, where \(t\) is the number of years after the year \(2010 .\) Predict the average population between the years 2010 and \(2060 .\)

Find each indefinite integral. \(\int \frac{4 x^{4}+4 x^{2}-x}{x} d x\)

Use a definite integral to find the area under each curve between the given \(x\) -values. For Exercises \(19-24\) also make a sketch of the curve showing the region. $$ f(x)=9-3 \sqrt{x} \text { from } x=0 \text { to } x=9 $$

Find each indefinite integral by the substitution method or state that it cannot be found by our substitution formulas. $$ \int \frac{x^{3}+x^{2}}{3 x^{4}+4 x^{3}} d x $$

Find each indefinite integral. \(\int \frac{x e^{x}+1}{x} d x\)

33-38. Find the derivative of each function. $$ \left(x^{5}-3 x^{3}+x-1\right)^{4} $$

Find each indefinite integral. \(\int(x-2)(x+4) d x\)

Use a definite integral to find the area under each curve between the given \(x\) -values. For Exercises \(19-24\) also make a sketch of the curve showing the region. $$ f(x)=\frac{1}{x} \text { from } x=1 \text { to } x=5 $$

A deposit of \(\$ 1000\) at \(5 \%\) interest compounded continuously will grow to \(V(t)=1000 e^{0.05 t}\) dollars after \(t\) years. Find the average value during the first 40 years (that is, from time 0 to time 40 ).

Find each indefinite integral by the substitution method or state that it cannot be found by our substitution formulas. $$ \int \frac{x^{2}-x}{2 x^{3}-3 x^{2}} d x $$

Find each indefinite integral. \(\int \frac{1}{x}\left(1-x e^{x}\right) d x\)

Find the derivative of each function. $$ \left(x^{4}-2 x^{2}-x+1\right)^{5} $$

A colony of bacteria is of size \(S(t)=300 e^{0.1 t}\) after \(t\) hours. Find the average size during the first 12 hours (that is, from time 0 to time 12 ).

Find each indefinite integral. \(\int(x+5)(x-3) d x\)

Use a definite integral to find the area under each curve between the given \(x\) -values. For Exercises \(19-24\) also make a sketch of the curve showing the region. $$ f(x)=\frac{1}{x} \text { from } x=e \text { to } x=e^{3} $$

Find each indefinite integral by the substitution method or state that it cannot be found by our substitution formulas. $$ \int \frac{x^{3}+x^{2}}{\left(3 x^{4}+4 x^{3}\right)^{2}} d x $$

35-40. Find each indefinite integral. [Hint: Use some algebra first. \(\int \frac{(x+1)^{2}}{x} d x\)

Revenues at 3D Systems are predicted to be \(14.2 x^{2}+120 x+474\) million dollars per year, where \(x\) is the number of years since \(2013 .\) Predict the average annual revenue from 2013 to $2023 .

Find each indefinite integral. \(\int(r-1)(r+1) d r\)

Use a definite integral to find the area under each curve between the given \(x\) -values. For Exercises \(19-24\) also make a sketch of the curve showing the region. $$ f(x)=x^{-1}+x^{2} \text { from } x=1 \text { to } x=2 $$

Find each indefinite integral by the substitution method or state that it cannot be found by our substitution formulas. $$ \int \frac{x^{2}-x}{\left(2 x^{3}-3 x^{2}\right)^{3}} d x $$

Find each indefinite integral. [Hint: Use some algebra first. \(\int \frac{(x-1)^{2}}{x} d x\)

Find the derivative of each function. $$ \ln \left(x^{3}-1\right) $$

Amazon's annual revenue is predicted to be \(1.6 x^{2}+17.4 x+74\) billion dollars, where \(x\) is the number of years since \(2013 .\) Predict Amazon's average revenue between 2013 and \(2023 .\)

Find each indefinite integral. \(\int(3 s+1)(3 s-1) d s\)

Use a definite integral to find the area under each curve between the given \(x\) -values. For Exercises \(19-24\) also make a sketch of the curve showing the region. $$ f(x)=6 e^{2 x} \text { from } x=0 \text { to } x=2 $$

Find each indefinite integral by the substitution method or state that it cannot be found by our substitution formulas. $$ \int \frac{x}{1-x^{2}} d x $$

Find each indefinite integral. [Hint: Use some algebra first. \(\int \frac{(t-1)(t+3)}{t^{2}} d t\)

Find the derivative of each function. $$ e^{x^{3}} $$

Find the area between the curve \(y=x^{2}+1\) and the line \(y=2 x-1\) (shown below) from \(x=0\) to \(x=3\) .

Find each indefinite integral. \(\int \frac{x^{2}-1}{x+1} d x\)

Use a definite integral to find the area under each curve between the given \(x\) -values. For Exercises \(19-24\) also make a sketch of the curve showing the region. $$ f(x)=2 e^{x} \text { from } x=0 \text { to } x=\ln 3 $$

Find each indefinite integral by the substitution method or state that it cannot be found by our substitution formulas. $$ \int \frac{1}{1-x} d x $$