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Problem 8.PE.87
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ cotx·csc³x dx
Problem 8.PE.97
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ θ·cos(2θ + 1) dθ
Problem 8.PE.105
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ (z + 1) / [z²(z² + 4)] dz
Problem 8.PE.123
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
123. ∫ √x * √(1 + √x) dx
Problem 8.PE.42
Evaluate the integrals in Exercises 37–44.
∫ sec²(θ) sin³(θ) dθ
Problem 8.PE.93
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ (2 − cosx + sinx) / sin²x dx
Problem 8.PE.107
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ t dt / √(9 − 4t²)
Problem 8.PE.73
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ x·sec²x dx
Problem 8.PE.67
Which of the improper integrals in Exercises 63–68 converge and which diverge?
∫ from −∞ to ∞ of (2 / (e^x + e^(−x))) dx
Problem 8.PE.77
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫₋₁⁰ e^x / (e^x + e^(−x)) dx
Problem 8.PE.10
Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [x / (x² + 4x + 3)] dx
Problem 8.PE.48
You are planning to use Simpson’s Rule to estimate the value of the integral Estimate ∫ from 1 to 2 of f(x) dx with an error magnitude less than 10⁻⁵ using Simpson’s Rule.
You have determined that |f⁽⁴⁾(x)| ≤ 3 throughout the interval of integration. How many subintervals should you use to ensure the required accuracy?
(Remember that for Simpson’s Rule the number of subintervals must be even.)
Problem 8.PE.57
Evaluate the improper integrals in Exercises 53–62.
∫ from 3 to ∞ of (2 / (u² − 2u)) du
Problem 8.PE.117
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
117. ∫ dr / (1 + √r)
Problem 8.PE.61
Evaluate the improper integrals in Exercises 53–62.
∫ from −∞ to ∞ of (1 / (4x² + 9)) dx
Problem 8.PE.109
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ e^t dt / (e^(2t) + 3e^t + 2)
Problem 8.PE.22
Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [(x³ + 1) / (x³ − x)] dx
Problem 8.PE.14
Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [cos(θ) / (sin²(θ) + sin(θ) − 6)] dθ
Problem 8.PE.59
Evaluate the improper integrals in Exercises 53–62.
∫ from 0 to ∞ of (x² * e^(−x)) dx
Problem 8.PE.125
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
125. ∫ dx / (√x * √(1 + x))
Problem 8.PE.115
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ (sin5t) dt / [1 + (cos5t)²]
Problem 8.PE.6
Evaluate the integrals in Exercises 1–8 using integration by parts.
∫ x² sin(1 − x) dx
Problem 8.PE.91
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ √(2x − x²) dx
Problem 8.PE.28
Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [1 / √(e^s + 1)] ds
Problem 8.PE.95
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ 9 dv / (81 − v⁴)
Problem 8.PE.29b
Evaluate the integrals in Exercises 29–32 (b) using a trigonometric substitution.
∫ [y / √(16 − y²)] dy
Problem 8.PE.24
Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [(2x³ + x² − 21x + 24) / (x² + 2x − 8)] dx
Problem 8.PE.111
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫₁^∞ (lny) / y³ dy
Problem 8.PE.50
Heat capacity of a gas
Heat capacity
C_v
is the amount of heat required to raise the temperature of a given mass of gas with constant volume by 1°C, measured in units of cal/deg-mol (calories per degree gram molecular weight).
The heat capacity of oxygen depends on its temperature T and satisfies the formula
C_v = 8.27 + 10^(-5) * (26T − 1.87T²)
Use Simpson’s Rule to find the average value of C_v and the temperature at which it is attained for
20°C ≤ T ≤ 675°C.
Problem 8.PE.69
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ x·e^(2x) dx