Chapter 6

$$ \int t e^{1-t} d t $$

$$ \int(5+3 x) e^{-x / 2} d x $$

$$ \int x \sqrt{2 x+3} d x $$

$$ \int_{-9}^{-1} \frac{y d y}{\sqrt{4-5 y}} $$

$$ \int_{1}^{4} \frac{\ln \sqrt{s}}{\sqrt{s}} d s $$

$$ \int(\ln x)^{2} d x $$

$$ \int_{-2}^{1}(2 x+1)(x+3)^{3 / 2} d x $$

$$ \int \frac{w^{3}}{\sqrt{1+w^{2}}} d w $$

$$ \int x^{3} \sqrt{3 x^{2}+2} d x $$

$$ \int_{0}^{1} \frac{x+2}{e^{3 x}} d x $$

Solve the given initial value problem using either integration by parts or a formula from Table 6.1. Note that Exercises 19 and 20 involve separable differential equations. $$ \frac{d y}{d x}=\frac{e^{y}}{x y}, \text { where } y=0 \text { when } x=1 $$

Solve the given initial value problem using either integration by parts or a formula from Table 6.1. Note that Exercises 19 and 20 involve separable differential equations. $$ \frac{d y}{d x}=\frac{x y}{3+x}, \text { where } y=1 \text { when } x=1 $$

Either evaluate the given improper integral or show that it diverges. $$ \int_{0}^{+\infty} \frac{1}{\sqrt[3]{1+2 x}} d x $$

Either evaluate the given improper integral or show that it diverges. $$ \int_{0}^{+\infty}(1+2 x)^{-3 / 2} d x $$

Either evaluate the given improper integral or show that it diverges. $$ \int_{0}^{+\infty} \frac{3 t}{t^{2}+1} d t $$

Either evaluate the given improper integral or show that it diverges. $$ \int_{0}^{+\infty} 3 e^{-5 x} d x $$

Either evaluate the given improper integral or show that it diverges. $$ \int_{0}^{+\infty} x e^{-2 x} d x $$

Either evaluate the given improper integral or show that it diverges. $$ \int_{0}^{+\infty} 2 x^{2} e^{-x^{3}} d x $$

Either evaluate the given improper integral or show that it diverges. $$ \int_{2}^{+\infty} \frac{1}{t(\ln t)^{2}} d t $$

Either evaluate the given improper integral or show that it diverges. $$ \int_{1}^{+\infty} \frac{\ln x}{\sqrt{x}} d x $$

Either evaluate the given improper integral or show that it diverges. $$ \int_{-\infty}^{-1} x e^{x+1} d x $$

Either evaluate the given improper integral or show that it diverges. $$ \int_{-\infty}^{\infty} x^{3} e^{-x^{2}} d x $$

Either evaluate the given improper integral or show that it diverges. $$ \int_{-\infty}^{\infty}\left(e^{x}+e^{-x}\right) d x $$

It is estimated that \(t\) years from now, a certain investment will be generating income at the rate of \(f(t)=8,000+400 t\) dollars per year. If the income is generated in perpetuity and the prevailing annual interest rate remains fixed at \(5 \%\) compounded continuously, find the present value of the investment.

PRODUCTION After \(t\) hours on the job, a factory worker can produce \(100 t e^{-0.5 t}\) units per hour. How many units does a worker who arrives on the job at 8:00 A.M. produce between 10:00 A.M. and noon?

DEMOGRAPHICS Demographic studies indicate that the fraction of the residents that will remain in a certain city for at least \(t\) years is \(f(t)=e^{-t / 20}\). The current population of the city is 100,000 , and it is estimated that \(t\) years from now, new people will be arriving at the rate of \(100 t\) people per year. If this estimate is correct, what will happen to the population of the city in the long run?

SUBSCRIPTION GROWTH The publishers of a national magazine have found that the fraction of subscribers who remain subscribers for at least \(t\) years is \(f(t)=e^{-t / 10}\). Currently, the magazine has 20,000 subscribers and it is estimated that new subscriptions will be sold at the rate of 1,000 per year. Approximately how many subscribers will the magazine have in the long run?

NUCLEAR WASTE After \(t\) years of operation, a certain nuclear power plant produces radioactive waste at the rate of \(R(t)\) pounds per year, where $$ R(t)=300-200 e^{-0.03 t} $$ The waste decays exponentially at the rate of \(2 \%\) per year. How much radioactive waste from the plant will be present in the long run?

PSYCHOLOGICAL TESTING In a psychological experiment, it is found that the proportion of participants who require more than \(t\) minutes to finish a particular task is given by $$ \int_{t}^{+\infty} 0.07 e^{-0.07 u} d u $$ a. Find the proportion of participants who require more than 5 minutes to finish the task. b. What proportion requires between 10 and 15 minutes to finish?

Approximate the given integral and estimate the error with the specified number of subintervals using: (a) The trapezoidal rule. (b) Simpson's rule. $$ \int_{1}^{3} \frac{1}{x} d x ; n=10 $$

Approximate the given integral and estimate the error with the specified number of subintervals using: (a) The trapezoidal rule. (b) Simpson's rule. $$ \int_{1}^{2} \frac{e^{x}}{x} d x ; n=10 $$

Approximate the given integral and estimate the error with the specified number of subintervals using: (a) The trapezoidal rule. (b) Simpson's rule. $$ \int_{1}^{2} x e^{1 / x} d x, n=8 $$

Determine how many subintervals are required to guarantee accuracy to within \(0.00005\) of the actual value of the given integral using: (a) The trapezoidal rule. (b) Simpson's rule. $$ \int_{0.5}^{1} e^{-1.1 x} d x $$

TOTAL COST FROM MARGINAL COST A manufacturer determines that the marginal cost of producing \(q\) units of a particular commodity is \(C^{\prime}(q)=\sqrt{q} e^{0.01 q}\) dollars per unit. a. Express the total cost of producing the first 8 units as a definite integral. b. Estimate the value of the total cost integral in part (a) using the trapezoidal rule with \(n=8\) subintervals.

Use the graphing utility of your calculator to draw the graphs of the curves \(y=-x^{3}-2 x^{2}+5 x-2\) and \(y=x \ln x\) for \(x>0\) on the same screen. Use ZOOM and TRACE or some other feature of your calculator to find where the curves intersect, and then compute the area of the region bounded by the curves.

Use the numeric integration feature of your calculator to compute $$ I(N)=\int_{0}^{N} \frac{1}{\sqrt{\pi}} e^{-x^{2}} d x $$ for \(N=1,10,50\). Based on your results, do you think the improper integral $$ \int_{0}^{+\infty} \frac{1}{\sqrt{\pi}} e^{-x^{2}} d x $$ converges? If so, to what value?

Use the numeric integration feature of your calculator to compute $$ I(N)=\int_{1}^{N} \frac{\ln (x+1)}{x} d x $$ for \(N=10,100,1,000,10,000\). Based on your results, do you think the improper integral $$ \int_{1}^{+\infty} \frac{\ln (x+1)}{x} d x $$ converges? If so, to what value?