Exercises

1. Decide whether each of the following integrals converges or diverges. If it converges, find its value.

   a.   `int_1^oo 1/x^4 dx`

   b.   `int_2^oo x^(-1//7) dx`

   c.   `int_2^oo x^(-7) dx

2. Why does it take two steps to evaluate an improper integral? In particular, what is it about the process of definite integration that requires an interval of finite length?
3. Find the cumulative failure fraction function `Ftext[(]t text[)]`, where time `t` is measured in days, for light bulbs with each of the following expected lifetimes.

   a.   1000 hours

   b.   750 hours

   c.   350 hours

4. 1. How big does `x` have to be in order that `ln x>100`?
   2. How big does `x` have to be in order that `ln x>1text[,]000text[,]000`? ("My calculator won't do that" is not a useful answer. Your computer algebra system will do it, if you ask the right question. Outthink your calculator or computer! Or look at this question the right way, and you won't need any help from a tool.)
   3. How do you know that `ln x rarr oo` as `x rarr oo`?
   4. How do you know that `int_2^oo 1/x dx` diverges?
      
5. 1. How big does `x` have to be in order that `ln(ln x)>5`?
   2. How big does `x` have to be in order that `ln(ln x)>100`?
   3. How do you know that `ln(ln x) rarr oo` as `x rarr oo`?
   4. How do you know that `int_2^oo 1/(x ln x) dx` diverges?
      
6. 1. Find all numbers `p` for which `int_1^oo 1/x^p dx` converges.
   2. For each such `p`, find the value of the integral in terms of `p`.
7. Evaluate each of the following integrals. Exact answers are preferable to approximations.

   a.   `int_3^oo x e^(-2x) dx`

   b.   `int_1^oo 1/(xtext[(]x+1text[)]) dx`

   c.   `int_1^oo 1/(x^2+1) dx

8. Guess each of the following limiting values. You may use a computer or calculator.

   a.  `lim_{x\to oo} x^(1//x)`	b.  `lim_{x\to oo} x sin 1/x`	c.  `lim_{x\to oo} (1+1/x)^x`
   d.  `lim_{x\to oo} (ln x)/x`	e.  `lim_{x\to oo} x^2/2^x`	f.  `lim_{x\to oo} x^10/10^x`
   g.  `lim_{x\to oo} (sqrt(1+x^2)-1)/x`	h.  `lim_{x\to oo} ln((sqrt(1+x^2)-1)/x)`	i.  `lim_{x\to oo} (1-1/x)^x`

9. 1. What is the probability that a light bulb with failure density function `ftext[(]t text[)]=r e^(-rt)` burns out before its expected lifetime?
   2. What is the probability that it lasts longer than its expected lifetime?
10. The median lifetime of a light bulb with failure density function `ftext[(]t text[)]=r e^(-rt)` is the time `T` such that the probability of failure before `t=T` is `1//2`. Find `T`. Is the median lifetime shorter, longer, or the same as the expected lifetime?
11. For a function `gtext[(]x text[)]` that is continuous for all `x`, we defined

    `int_-oo^oo gtext[(]x text[)] dx=int_-oo^0 gtext[(]x text[)] dx+int_0^oo gtext[(]x text[)] dx`,

    with the understanding that the two-sided improper integral has a value only if both integrals on the right converge. Explain why splitting at `0` is merely a matter of convenience — that is,

    `int_-oo^oo gtext[(]x text[)] dx=int_-oo^a gtext[(]x text[)] dx+int_a^oo gtext[(]x text[)] dx`

    for any number `a` with exactly the same meaning. You have two things to show:
    
    
    1. If the two-sided integral converges by either definition, it converges by the other as well.
    2. If the two-sided integral converges, then both definitions give the same value.
12. In this exercise we explore why it is necessary to split a two-sided improper integral into two parts, with separate evaluations of limiting values, in order to get right answers.
    1. Graph the function `ftext[(]x text[)]=x/(1+x^2)`.
    2. Show that `int_0^oo x/(1+x^2) dx` does not converge. (Hint: See Exercise 4.)
    3. For any positive number `T`, what is the value of `int_-T^T x/(1+x^2) dx`? Use part (a) to answer this graphically. Use your computer or calculator (for selected values of `T`) to answer numerically. Use the antiderivative you calculated in part (b) to answer symbolically.
    4. Find `lim_{T\to oo} int_-T^T x/(1+x^2) dx`. Explain why this limiting value is not the value of `int_-oo^oo x/(1+x^2) dx`.
13. 1. Suppose `ftext[(]x text[)]` is an even function that is continuous for all `x` and for which `int_0^oo ftext[(]x text[)] dx` converges. Explain why

       `int_-oo^oo ftext[(]x text[)] dx=2 int_0^oo ftext[(]x text[)] dx`.

    2. Use part (a) to evaluate `int_-oo^oo 1/(1+x^2) dx`. Compare to Example 5.
14. 1. Suppose `ftext[(]x text[)]` is an odd function that is continuous for all `x` and for which `int_0^oo ftext[(]x text[)] dx` converges. Explain why `int_-oo^oo ftext[(]x text[)] dx=0`.
    2. Use part (a) to evaluate `int_-oo^oo x e^(-x^2) dx`. Be careful: You have to show that all the conditions on the function are satisfied.

When an improper integral cannot be evaluated directly, it may be possible to determine whether it converges or diverges by comparing it to another integral whose convergence or divergence is known. The remaining exercises are related to this idea.

15. 1. Explain why `1/(1+x^(3//2))<1/x^(3//2)` for `x>=1`.
    2. Explain why `int_1^oo 1/(1+x^(3//2)) dx` converges. (See Exercise 6.)
    3. Estimate the integral in part (b).
16. 1. Explain why `x/(1+x^(3//2))>x/(1+x^2)` for `x>=1`.
    2. Explain why `int_1^oo x/(1+x^(3//2)) dx` diverges. (See Exercise 12.)
17. 1. Invent a test for convergence or divergence of integrals of the form `int_a^oo ftext[(]x text[)] dx` and `int_a^oo gtext[(]x text[)] dx` where `f` and `g` are assumed to be continuous and nonnegative for `x>=a`.
    2. Give a geometric argument to justify your comparison test for convergence or divergence.
       
       
    Problem adapted from Calculus Problems for a New Century, edited by Robert Fraga, MAA Notes No. 28, 1993.
18. Decide whether each of the following integrals converges or diverges (see Exercise 17).

    a.   `int_1^oo 1/(1+x^4) dx`	b.   `int_2^oo x/sqrt(1+x^3) dx`	c.   `int_1^oo 1/(x sqrt(1+x^2)) dx`
    d.   `int_0^oo e^(-x^3) dx`	e.   `int_2^oo 1/(text[(]x+2text[)]^2) dx`	f.   `int_0^oo 1/(1+e^x) dx`

19. For each of the following integrals that converges (see Exercise 18), use your computer or calculator to estimate the value.

    a.   `int_1^oo 1/(1+x^4) dx`	b.   `int_2^oo x/sqrt(1+x^3) dx`	c.   `int_1^oo 1/(x sqrt(1+x^2)) dx`
    d.   `int_0^oo e^(-x^3) dx`	e.   `int_2^oo 1/(text[(]x+2text[)]^2) dx`	f.   `int_0^oo 1/(1+e^x) dx`

20. 1. Evaluate `lim_{x\to oo} (sin x)/x^2`.
    2. The function `ftext[(]x text[)]= (sin x)/x^2` has both positive and negative values. Expand on the idea in Exercise 17 to show that `int_1^oo ftext[(]x text[)] dx` converges. (Your computer algebra system can help you evaluate this integral. Your calculator probably can't.)
21. 1. Explain carefully why `int_1^oo 1/(x sqrt(1+x^2)) dx` converges.
    2. Evaluate the integral in part (a).