Chapter 6

In Exercises \(25-46,\) use substitution to evaluate the integral. $$\int \frac{d x}{\sqrt{5 x+8}}$$

In Exercises \(41-44,\) use Euler's Method with increments of \(\Delta x=0.1\) to approximate the value of \(y\) when \(x=1.3 .\) \(\frac{d y}{d x}=2 x-y\) and \(y=0\) when \(x=1\)

John Napier's Answer John Napier \((1550-1617),\) the Scottish laird who invented logarithms, was the first person to answer the question, "What happens if you invest an amount of money at 100\(\%\) yearly interest, compounded continuously?" (a) Writing to Learn What does happen? Explain. (b) How long does it take to triple your money? (c) Writing to Learn How much can you earn in a year?

In Exercises \(43-46\) , evaluate the integral by using a substitution prior to integration by parts. $$\int e^{\sqrt{3 x+9}} d x$$

In Exercises \(45-48\) , use Euler's Method with increments of \(\Delta x=-0.1\) to approximate the value of \(y\) when \(x=1.7\) \(\frac{d y}{d x}=2-x\) and \(y=1\) when \(x=2\)

Extinct Populations One theory states that if the size of a population falls below a minimum \(m,\) the population will become extinct. This condition leads to the extended logistic differential equation \(\frac{d P}{d t}=k P\left(1-\frac{P}{M}\right)\left(1-\frac{m}{P}\right)\) with \(k>0\) the proportionality constant and \(M\) the population maximum. (a) Show that dP&dt is positive for m < P < M and negative if P M. (b) Let \(m=100, \)M = 1200, and assume that m < P < M. Show that the differential equation can be rewritten in the form \(\left[\frac{1}{1200-P}+\frac{1}{P-100}\right] \frac{d P}{d t}=\frac{11}{12} k\) Use a procedure similar to that used in Example 5 in Section 6.5 to solve this differential equation. (c) Find the solution to part (b) that satisfies \(P(0)=300\) . (d) Superimpose the graph of the solution in part (c) with \(k=0.1\) on a slope field of the differential equation. (e) Solve the general extended differential equation with the restriction m

(a) What annual rate of interest, compounded continuously for 100 years, would have multiplied Benjamin Franklin's original capital by \(90\) ? (b) In Benjamin Franklin's estimate that the original 1000 pounds would grow to \(131,000\) in 100 years, he was using an annual rate of 5\(\%\) and compounding once each year. What rate of interest per year when compounded continuously for 100 years would multiply the original amount by 131 ?

In Exercises \(43-46\) , evaluate the integral by using a substitution prior to integration by parts. $$\int x^{7} e^{x^{2}} d x$$

\(\int \sec x d x \quad\) (Hint: Multiply the integrand by \(\frac{\sec x+\tan x}{\sec x+\tan x}\) and then use a substitution to integrate the result.)

Integral Tables Antiderivatives of various generic functions can be found as formulas in integral tables. See if you can derive the formulas that would appear in an integral table for the fol- lowing functions. (Here, \(a\) is an arbitrary constant.) See below. (a) \(\int \frac{d x}{a^{2}+x^{2}} \quad\) (b) \(\int \frac{d x}{a^{2}-x^{2}} \quad\) (c) \(\int \frac{d x}{(a+x)^{2}}\)