12. Techniques of Integration

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.

∫₀¹ (4r dr) / √(1 − r⁴)

62. Electronic Chips Suppose the probability that a particular computer chip fails after a hours of operation is 0.00005 ∫(from a to ∞) e^(-0.00005t) dt.

a. Find the probability that the computer chip fails after 15,000 hr of operation.

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.

∫ from 2 to ∞ of (dx / √(x² - 1))

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.

∫ from 1 to ∞ of ((1 / (e^x - 2^x)) dx)

In Exercises 35–68, use integration, the Direct Comparison Test, or the Limit Comparison Test to test the integrals for convergence. If more than one method applies, use whatever method you prefer.

∫ from e to e^e of (ln(ln x) dx)

7–58. Improper integrals Evaluate the following integrals or state that they diverge.

44. ∫ (from 0 to ln 3) eʸ/(eʸ-1)⁷ᐟ³ dy

Exercises 83–86 are about the infinite region in the first quadrant between the curve y = e^(-x) and the x-axis.

86. Find the volume of the solid generated by revolving the region about the x-axis.

Which of the improper integrals in Exercises 63–68 converge and which diverge?

∫ from 1 to ∞ of ((ln z) / z) dz

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.

∫₋∞^∞ 2x e^(−x²) dx

95–98. {Use of Tech} Numerical methods Use numerical methods or a calculator to approximate the following integrals as closely as possible. The exact value of each integral is given.

96. ∫(from 0 to ∞) (sin²x)/x² dx = π/2

7–58. Improper integrals Evaluate the following integrals or state that they diverge.

53. ∫ (from 0 to 1) ln x dx

59. Perpetual Annuity

Imagine that today you deposit \(B in a savings account that earns interest at a rate of *p*% per year compounded continuously (see Section 7.2). The goal is to draw an income of \)I per year from the account forever. The amount of money that must be deposited is:

B = I × ∫(from 0 to ∞) e^(-rt) dt

where r = p/100.

Suppose you find an account that earns 12% interest annually, and you wish to have an income from the account of \$5000 per year. How much must you deposit today?

For each x > 0, let G(x) = ∫(from 0 to x) e^(-xt) dt. Prove that xG(x) = 1 for each x > 0.

7–58. Improper integrals Evaluate the following integrals or state that they diverge.

33. ∫ (from 2 to ∞) 1/(y ln y) dy

Evaluate the improper integrals in Exercises 53–62.

∫ from 0 to ∞ of (x² * e^(−x)) dx