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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.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.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.31b
Evaluate the integrals in Exercises 29–32 (b) using a trigonometric substitution.
∫ [x / √(4 − x²)] dx
Problem 8.PE.133
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
133. ∫ (sin²x) / (1 + sin²x) dx
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
Problem 8.PE.44
Evaluate the integrals in Exercises 37–44.
∫ eᵗ √[tan²(eᵗ) + 1] dt
Problem 8.PE.75
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ sinx·cos²x dx
Problem 8.PE.113
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^(ln√x) dx
Problem 8.PE.127
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
127. ∫ (ln x) / (x + x ln x) dx
Problem 8.PE.99
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ sin(2θ) dθ / (1 + cos(2θ))²
Problem 8.PE.131
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
131. ∫ dx / (x√(1 − x⁴))
Problem 8.PE.103
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫ dy / (y² − 2y + 2)
Problem 8.PE.8
Evaluate the integrals in Exercises 1–8 using integration by parts.
∫ x sin(x) cos(x) dx
Problem 8.PE.121
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
121. ∫ (1 + x²) / (1 + x³) dx
Problem 8.PE.119
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
119. ∫ x³ / (1 + x²) dx
Problem 8.PE.30b
Evaluate the integrals in Exercises 29–32 (b) using a trigonometric substitution.
∫ [x / √(4 + x²)] dx
Problem 8.PE.46
A brief calculation shows that if 0 ≤ x ≤ 1, then the second derivative of
f(x) = √(1 + x⁴)
lies between 0 and 8.
Based on this, about how many subdivisions would you need to estimate the integral of f from 0 to 1
with an error no greater than 10⁻³ in absolute value using the Trapezoidal Rule?
Problem 8.PE.26
Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [1 / (x (1 + ∛x))] dx
Problem 8.PE.81
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^x + e^(3x)) / e^(2x) dx
Problem 8.PE.55
Evaluate the improper integrals in Exercises 53–62.
∫ from 0 to 2 of (1 / (y − 1)^(2/3)) dy
Problem 8.PE.32b
Evaluate the integrals in Exercises 29–32 (b) using a trigonometric substitution.
∫ [t / √(4t² − 1)] dt
Problem 8.PE.2
Evaluate the integrals in Exercises 1–8 using integration by parts.
∫ x² ln(x) dx
Problem 8.PE.83
Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.
∫₀^(π/3) tan³x·sec²x dx
Problem 8.PE.63
Which of the improper integrals in Exercises 63–68 converge and which diverge?
∫ from 6 to ∞ of (1 / √(θ² + 1)) dθ
Problem 8.PE.38
Evaluate the integrals in Exercises 37–44.
∫ cos⁵(x) sin⁵(x) dx
Problem 8.PE.79
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 + 1) / (x⁴ − x³) dx
Problem 8.PE.16
Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.
∫ [(4x) / (x³ + 4x)] dx