Chapter 5

Find each indefinite integral. [Hint: Use some algebra first. \(\int \frac{(t+2)(t-4)}{t^{2}} d t\)

Find the derivative of each function. $$ e^{x^{4}} $$

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

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. $$ f(x)=e^{-x} \text { from } x=0 \text { to } x=1 $$

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

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

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

Suppose that for a demand function \(d(x)\) we have \(d(0)=1000 .\) Describe in everyday language what this means about the number 1000 .

Find the area between the curves \(y=e^{x}\) and \(y=e^{2 x} \quad\) (shown below) from \(x=0\) to \(x=2\). (Leave the answer in its exact form.)

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. $$ f(x)=e^{x / 2} \text { from } x=0 \text { to } x=2 $$

Find each indefinite integral. \(\int(t+1)^{3} d t\)

Find each indefinite integral by the substitution method or state that it cannot be found by our substitution formulas. $$ \int(5 x+9)^{9} d x $$

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

Suppose that for a demand function \(d(x)\) we have \(d(1000)=0 .\) Describe in everyday language what this means about the number \(1000 .\)

Find the area between the curves \(y=e^{x}\) and \(y=e^{-x}\) (shown below) from \(x=0\) to \(x=1\). (Leave the answer in its exact form.)

Find each indefinite integral. \(\int(t-1)^{3} d t\)

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. $$ f(x)=e^{x / 3} \text { from } x=0 \text { to } x=3 $$

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

41-42. A flu epidemic hits a college community, beginning with five cases on day \(t=0\). The rate of growth of the epidemic (new cases per day) is given by the following function \(r(t),\) where \(t\) is the number of days since the epidemic began. a. Find a formula for the total number of cases of flu in the first \(t\) days. b. Use your answer to part (a) to find the total number of cases in the first 20 days. \(r(t)=18 e^{0.05 t}\)

Sketch each parabola and line on the same graph and find the area between them from \(x=0\) to \(x=3\). \(y=x^{2}+4\) and \(y=2 x+1\)

If \(d(x)\) is the demand function for a product, what would it mean about the product if \(d(0)=0 ?\)

For each function: a. Integrate ("by hand") to find the area under the curve between the given \(x\) -values. b. Verify your answer to part (a) by having your calculator graph the function and find the area (using a command like FnInt or \(\int \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}\) ). $$ f(x)=12-3 x^{2} \text { from } x=1 \text { to } x=2 $$

a. Verify that \(\int x^{2} d x=\frac{1}{3} x^{3}+C\). b. Graph the five functions \(\frac{1}{3} x^{3}-2, \frac{1}{3} x^{3}-1\), \(\frac{1}{3} x^{3}, \frac{1}{3} x^{3}+1,\) and \(\frac{1}{3} x^{3}+2\) (the solutions for five different values of \(C\) ) on the window [-3,3] by \([-5,5] .\) Use TRACE to see how the constant shifts the curve vertically. c. Find the slopes (using NDERIV or \(d y / d x\) ) of several of the curves at a particular \(x\) -value and check that in each case the slope is the square of the \(x\) -value. This verifies that the derivative of each curve is \(x^{2}\), and so each is an integral of \(x^{2}\).

Find each indefinite integral by the substitution method or state that it cannot be found by our substitution formulas. $$ \int \frac{e^{3 x}}{e^{3 x}-1} d x $$

A flu epidemic hits a college community, beginning with five cases on day \(t=0\). The rate of growth of the epidemic (new cases per day) is given by the following function \(r(t),\) where \(t\) is the number of days since the epidemic began. a. Find a formula for the total number of cases of flu in the first \(t\) days. b. Use your answer to part (a) to find the total number of cases in the first 20 days. \(r(t)=20 e^{0.04 t}\)

Sketch each parabola and line on the same graph and find the area between them from \(x=0\) to \(x=3\). \(y=x^{2}+5\) and \(y=2 x+3\)

Should demand curves slope upward or downward? Why?

For each function: a. Integrate ("by hand") to find the area under the curve between the given \(x\) -values. b. Verify your answer to part (a) by having your calculator graph the function and find the area (using a command like FnInt or \(\int \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}\) ). $$ f(x)=9 x^{2}-6 x+1 \text { from } x=1 \text { to } x=2 $$

a. Graph the five functions \(\ln x-2, \ln x-1, \ln x\) \(\ln x+1,\) and \(\ln x+2\) on the window [0,4] by [-3,3] b. Find the slope (using NDERIV or \(d y / d x\) ) of several of the curves at a particular \(x\) -value and check that in each case the slope is the reciprocal of the \(x\) -value. This suggests that the derivative of each function is \(1 / x\) c. Based on part (b), conjecture what is the indefinite integral of the function \(1 / x\) (for \(x>0\) ).

Find each indefinite integral by the substitution method or state that it cannot be found by our substitution formulas. $$ \int \frac{\ln x}{x} d x \quad[\text { Hint: Let } u=\ln x .] $$

In an effort to reduce its inventory, a warehouse runs a sale on its least popular Blu-ray discs. The sales rate (discs sold per day) on day \(t\) of the sale is predicted to be \(50 / t\) (for \(t \geq 1),\) where \(t=1\) corresponds to the beginning of the sale, at which time none of the inventory of 200 discs had been sold. a. Find a formula for the total number of discs sold up to day \(t\) b. Will the store have sold its inventory of 200 discs by day \(t=30 ?\)

Should supply curves slope upward or downward? Why?

Sketch each parabola and line on the same graph and find the area between them from \(x=0\) to \(x=3\). \(y=3 x^{2}-3\) and \(y=2 x+5\)

For each function: a. Integrate ("by hand") to find the area under the curve between the given \(x\) -values. b. Verify your answer to part (a) by having your calculator graph the function and find the area (using a command like FnInt or \(\int \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}\) ). $$ f(x)=\frac{1}{x^{3}} \text { from } x=1 \text { to } x=4 $$

Find each indefinite integral by the substitution method or state that it cannot be found by our substitution formulas. $$ \int \frac{(\ln x)^{2}}{x} d x \quad[\text { Hint: Let } u=\ln x .] $$

In an effort to make room for new inventory, a college bookstore runs a sale on its least popular mathematics books. The sales rate (books sold per day) on day \(t\) of the sale is predicted to be \(60 / t\) (for \(t \geq 1\) ), where \(t=1\) corresponds to the beginning of the sale, at which time none of the inventory of 350 books had been sold. a. Find a formula for the number of books sold up to day \(t\) b. Will the store have sold its inventory of 350 books by day \(t=30 ?\)

For a Lorenz curve \(L(x)\), what must be the values of \(L(0)\) and \(L(1) ?\)

Sketch each parabola and line on the same graph and find the area between them from \(x=0\) to \(x=3\). \(y=3 x^{2}-12\) and \(y=2 x-11\)

For each function: a. Integrate ("by hand") to find the area under the curve between the given \(x\) -values. b. Verify your answer to part (a) by having your calculator graph the function and find the area (using a command like FnInt or \(\int \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}\) ). $$ f(x)=\frac{1}{\sqrt[3]{x}} \text { from } x=8 \text { to } x=27 $$

Find each indefinite integral by the substitution method or state that it cannot be found by our substitution formulas. $$ \int \frac{e^{\sqrt{x}}}{\sqrt{x}} d x \quad[\text { Hint: Let } u=\sqrt{x} .] $$

World consumption of tin is running at the rate of \(342 e^{0.02 t}\) thousand metric tons per year, where \(t\) is measured in years and \(t=0\) corresponds to 2014 a. Find a formula for the total amount of tin that will be consumed within \(t\) years of 2014 b. When will the known world resources of 4900 thousand metric tons of tin be exhausted? [Tin is used mainly for coating steel (a "tin" can is actually a steel can with a thin protective coating of tin to prevent rust).

Find the area bounded by the given curves. \(y=x^{2}-1\) and \(y=2-2 x^{2}\)

For each function: a. Integrate ("by hand") to find the area under the curve between the given \(x\) -values. b. Verify your answer to part (a) by having your calculator graph the function and find the area (using a command like FnInt or \(\int \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}\) ). $$ f(x)=2 x+1+x^{-1} \text {from } x=1 \text { to } x=2 $$

Find each indefinite integral by the substitution method or state that it cannot be found by our substitution formulas. $$ \int \frac{e^{1 / x}}{x^{2}} d x \quad\left[\text { Hint: Let } u=\frac{1}{x} .\right] $$

World consumption of copper is running at the rate of \(18.8 e^{0.04 t}\) million metric tons per year, where \(t\) is measured in years and \(t=0\) corresponds to 2014 a. Find a formula for the total amount of copper that will be used within \(t\) years of 2014 . b. When will the known world resources of 680 million metric tons of copper be exhausted? Source: U.S. Geological Survey

Find the area bounded by the given curves. \(y=x^{2}-4\) and \(y=8-2 x^{2}\)

For each function: a. Integrate ("by hand") to find the area under the curve between the given \(x\) -values. b. Verify your answer to part (a) by having your calculator graph the function and find the area (using a command like FnInt or \(\int \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}\) ). $$ f(x)=e^{x} \text { from } x=0 \text { to } x=3 $$

47-50. Find each integral. [Hint: Try some algebra.] $$ \int(x+1) x^{2} d x $$

The cost of maintaining a home generally increases as the home becomes older. Suppose that the maintenance costs increase at the rate of \(1800 e^{0.05 x}\) (dollars per year) when the home is \(x\) years old. a. Find a formula for the total maintenance cost during the first \(x\) years. (Total maintenance should be zero at \(x=0 .)\) b. Use your answer to part (a) to find the total maintenance cost during the first 5 years.

Find the area bounded by the given curves. \(y=6 x^{2}-10 x-8\) and \(y=3 x^{2}+8 x-23\)