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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.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.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
Problem 8.PE.31b
Evaluate the integrals in Exercises 29–32 (b) using a trigonometric substitution.
∫ [x / √(4 − x²)] dx
Problem 8.PE.44
Evaluate the integrals in Exercises 37–44.
∫ eᵗ √[tan²(eᵗ) + 1] dt
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.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.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.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.8
Evaluate the integrals in Exercises 1–8 using integration by parts.
∫ x sin(x) cos(x) 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.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.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.32b
Evaluate the integrals in Exercises 29–32 (b) using a trigonometric substitution.
∫ [t / √(4t² − 1)] dt
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.55
Evaluate the improper integrals in Exercises 53–62.
∫ from 0 to 2 of (1 / (y − 1)^(2/3)) dy
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.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.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.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.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