Chapter 6

17-40. Evaluate each improper integral or state that it is divergent. $$ \int_{-\infty}^{0} \frac{x^{4}}{\left(x^{5}-1\right)^{2}} d x $$

Use integration by parts to find each integral. \(\int \ln x^{2} d x\)

Find each integral by using the integral table on the inside back cover. $$ \int x^{2} \sqrt{x^{6}+1} d x $$

17-40. Evaluate each improper integral or state that it is divergent. $$ \int_{-\infty}^{1} \frac{1}{2-x} d x $$

Use integration by parts to find each integral. \(\int x^{3} e^{x^{2}} d x \quad\left[\right.\)and use a substitution to find \(v\) from \(d v\).

17-40. Evaluate each improper integral or state that it is divergent. $$ \int_{-\infty}^{0} \frac{1}{1-x} d x $$

Use integration by parts to find each integral. \(\int x^{3}\left(x^{2}-1\right)^{6} d x\)

Find each integral by using the integral table on the inside back cover. $$ \int \frac{\sqrt{1-x^{6}}}{x} d x $$

17-40. Evaluate each improper integral or state that it is divergent. $$ \int_{-\infty}^{\infty} \frac{e^{x}}{\left(1+e^{x}\right)^{2}} d x $$

Find each integral by integration by parts or a substitution, as appropriate. a. \(\int x e^{x^{2}} d x\) b. \(\int \frac{(\ln x)^{3}}{x} d x\) c. \(\int x^{2} \ln 2 x d x\) d. \(\int \frac{e^{x}}{e^{x}+4} d x\)

Find each integral by using the integral table on the inside back cover. $$ \int \frac{e^{t}}{\left(e^{t}-1\right)\left(e^{t}+1\right)} d t $$

17-40. Evaluate each improper integral or state that it is divergent. $$ \int_{-\infty}^{\infty} \frac{e^{-x}}{\left(1+e^{-x}\right)^{3}} d x $$

Find each integral by integration by parts or a substitution, as appropriate. a. \(\int \sqrt{\ln x} \frac{1}{x} d x\) b. \(\int x^{2} e^{x^{3}} d x\) c. \(\int x^{7} \ln 3 x d x\) d. \(\int x e^{4 x} d x\)

Find each integral by using the integral table on the inside back cover. $$ \int \frac{e^{2 t}}{\left(e^{t}-1\right)\left(e^{t}+1\right)} d t $$

17-40. Evaluate each improper integral or state that it is divergent. $$ \int_{-\infty}^{\infty} \frac{e^{x}}{1+e^{x}} d x $$

Evaluate each definite integral using integration by parts. (Leave answers in exact form.) \(\int_{0}^{2} x e^{x} d x\)

Find each integral by using the integral table on the inside back cover. $$ \int x e^{x / 2} d x $$

17-40. Evaluate each improper integral or state that it is divergent. $$ \int_{-\infty}^{\infty} \frac{e^{-x}}{1+e^{-x}} d x $$

Evaluate each definite integral using integration by parts. (Leave answers in exact form.) \(\int_{0}^{3} x e^{x} d x\)

Find each integral by using the integral table on the inside back cover. $$ \int x e^{x / 2} d x $$

Use a graphing calculator to estimate the improper integrals \(\int_{0}^{\infty} e^{\sqrt{x}} d x\) and \(\int_{0}^{\infty} e^{-x^{2}} d x\) (if they converge) as follows: a. Define \(y_{1}\) to be the definite integral (using FnInt) of \(e^{\sqrt{x}}\) from 0 to \(x\). b. Define \(y_{2}\) to be the definite integral of \(e^{-x^{2}}\) from 0 to \(x\) c. \(y_{1}\) and \(y_{2}\) then give the areas under these curves out to any number \(x .\) Make a TABLE of values of \(y_{1}\) and \(y_{2}\) for \(x\) -values such as \(1,10,100,\) and \(500 .\) Which integral converges (and to what number, approximated to five decimal places) and which diverges?

Evaluate each definite integral using integration by parts. (Leave answers in exact form.) \(\int_{1}^{3} x^{2} \ln x d x\)

Find each integral by using the integral table on the inside back cover. $$ \int \frac{1}{e^{-x}+4} d x $$

Use a graphing calculator to estimate the mproper integrals \(\int_{0}^{\infty} \frac{1}{x^{2}+1} d x\) and \(\int_{0}^{\infty} \frac{1}{\sqrt{x}+1} d x\) (if they converge) as follows: Wefine \(y_{1}\) to be the definite integral (using FnInt) of \(\frac{1}{x^{2}+1}\) from 0 to \(x\). c. \(y_{1}\) and \(y_{2}\) then give the areas under these curves out to any number \(x\). Make a TABLE of values of \(y_{1}\) and \(y_{2}\) for \(x\) -values such as \(1,10,100,500,\) and b. Define \(y_{2}\) to be the definite integral of \(\frac{1}{\sqrt{x}+1}\) from 0 to \(x\)

Evaluate each definite integral using integration by parts. (Leave answers in exact form.) \(\int_{1}^{2} x \ln x d x\)

Find each integral by using the integral table on the inside back cover. $$ \int \frac{1}{\sqrt{e^{-x}+4}} d x $$

GENERAL: Permanent Endowments Find the size of the permanent endowment needed to generate an annual \(\$ 12,000\) forever at a continuous interest rate of \(6 \%\).

Evaluate each definite integral using integration by parts. (Leave answers in exact form.) \(\int_{0}^{2} z(z-2)^{4} d z\)

GENERAL: Permanent Endowments Show that the size of the permanent endowment needed to generate an annual \(C\) dollars forever at interest rate \(r\) compounded continuously is \(C / r\) dollars.

Evaluate each definite integral using integration by parts. (Leave answers in exact form.) \(\int_{0}^{4} 2(z-4)^{6} d z\)

For each definite integral: a. Evaluate it using the table of integrals on the inside back cover. (Leave answers in exact form.) b. Use a graphing calculator to verify your answer to part (a). $$ \int_{0}^{4} \frac{1}{\sqrt{x^{2}+9}} d x $$

a. Find the size of the permanent endowment needed to generate an annual \(\$ 1000\) forever at a continuous interest rate of \(10 \%\) b. At this same interest rate, the size of the fund needed to generate an annual \(\$ 1000\) for precisely 100 years is \(\int_{0}^{100} 1000 e^{-0.1 t} d t .\) Evaluate this integral (it is not an improper integral), approximating your answer using a calculator. c. Notice that the cost for the first 100 years is almost the same as the cost forever. This illustrates again the principle that in endowments, the short term is expensive, but eternity is cheap.

Evaluate each definite integral using integration by parts. (Leave answers in exact form.) \(\int_{0}^{\ln 4} t e^{t} d t\)

\(46-48 .\) BUSINESS: Capital Value of an Asset The capital value of an asset is defined as the present value of all future earnings. For an asset that may last indefinitely (such as real estate or a corporation), the capital value is $$ \left(\begin{array}{c} \text { Capital } \\ \text { value } \end{array}\right)=\int_{0}^{\infty} C(t) e^{-r t} d t $$ where \(C(t)\) is the income per year and \(r\) is the continuous interest rate. Find the capital value of a piece of property that will generate an annual income of \(C(t),\) for the function \(C(t)\) given below, at a continuous interest rate of \(5 \%\). $$ C(t)=8000 \text { dollars } $$

Evaluate each definite integral using integration by parts. (Leave answers in exact form.) \(\int_{1}^{e} \ln x d x\)

For each definite integral: a. Evaluate it using the table of integrals on the inside back cover. (Leave answers in exact form.) b. Use a graphing calculator to verify your answer to part (a). $$ \int_{2}^{4} \frac{1}{1-x^{2}} d x $$

\(46-48 .\) BUSINESS: Capital Value of an Asset The capital value of an asset is defined as the present value of all future earnings. For an asset that may last indefinitely (such as real estate or a corporation), the capital value is $$ \left(\begin{array}{c} \text { Capital } \\ \text { value } \end{array}\right)=\int_{0}^{\infty} C(t) e^{-r t} d t $$ where \(C(t)\) is the income per year and \(r\) is the continuous interest rate. Find the capital value of a piece of property that will generate an annual income of \(C(t),\) for the function \(C(t)\) given below, at a continuous interest rate of \(5 \%\). $$ C(t)=50 \sqrt{t} \text { thousand dollars } $$

Find in two different ways and check that your answers agree. \(\int x(x-2)^{5} d x\) a. Use integration by parts. b. Use the substitution \(u=x-2\) (so \(x\) is replaced by \(u+2\) ) and then multiply out the integrand.

\(46-48 .\) BUSINESS: Capital Value of an Asset The capital value of an asset is defined as the present value of all future earnings. For an asset that may last indefinitely (such as real estate or a corporation), the capital value is $$ \left(\begin{array}{c} \text { Capital } \\ \text { value } \end{array}\right)=\int_{0}^{\infty} C(t) e^{-r t} d t $$ where \(C(t)\) is the income per year and \(r\) is the continuous interest rate. Find the capital value of a piece of property that will generate an annual income of \(C(t),\) for the function \(C(t)\) given below, at a continuous interest rate of \(5 \%\). $$ C(t)=59 t^{0.1} \text { thousand dollars } $$

Find in two different ways and check that your answers agree. \(\int x(x+4)^{6} d x\) a. Use integration by parts. b. Use the substitution \(u=x+4\) (so \(x\) is replaced by \(u-4\) ) and then multiply out the integrand.

For each definite integral: a. Evaluate it using the table of integrals on the inside back cover. (Leave answers in exact form.) b. Use a graphing calculator to verify your answer to part (a). $$ \int_{3}^{4} \frac{1}{x \sqrt{25-x^{2}}} d x $$

BUSINESS: Oil Well Output An oil well is expected to produce oil at the rate of \(50 e^{-0.05 t}\) thousand barrels per month indefinitely, where \(t\) is the number of months that the well has been in operation. Find the total output over the lifetime of the well by integrating this rate from 0 to \(\infty\). [Note: The owner will shut down the well when production falls too low, but it is convenient to estimate the total output as if production continued forever.]

Derive each formula by using integration by parts on the left-hand side. (Assume \(n>0 .)\) \(\int x^{\mathrm{n}} e^{x} d x=x^{\mathrm{n}} e^{x}-n \int x^{n-1} e^{x} d x\)

\(49-56 .\) Find each integral by whatever means are necessary (either substitution or tables). $$ \int \frac{1}{2 x+6} d x $$

GENERAL: Duration of Telephone Calls The proportion of telephone calls that last longer than \(t\) minutes is approximately \(\int_{t}^{\infty} 0.3 e^{-0.3 s} d s .\) Use this formula to find the proportion of telephone calls that last longer than 4 minutes.

Derive each formula by using integration by parts on the left-hand side. (Assume \(n>0 .)\) \(\int(\ln x)^{n} d x=x(\ln x)^{n}-n \int(\ln x)^{n-1} d x\)

Find each integral by whatever means are necessary (either substitution or tables). $$ \int \frac{x}{x^{2}-4} d x $$

AREA Find the area between the curve \(y=1 / x^{3 / 2}\) and the \(x\) -axis from \(x=1\) to \(\infty\).

Find each integral by whatever means are necessary (either substitution or tables). $$ \int \frac{x}{2 x+6} d x $$

AREA Find the area between the curve \(y=e^{-4 x}\) and the \(x\) -axis from \(x=0\) to \(\infty\).