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문제 8.1.34
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ e^(z + eᶻ) dz
문제 8.8.44
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 0 to 1 of ((e^(-√x)) / √x dx)
문제 8.4.28
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ dx / (4 - x²)^(3/2) from 0 to 1
문제 8.5.4
Expand the quotients in Exercises 1–8 by partial fractions.
(2x + 2) / (x² - 2x + 1)
문제 8.1.45
Area: Find the area of the region bounded above by y = 2 cos x and below by y = sec x, −π/4 ≤ x ≤ π/4.
문제 8.1.8
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (2 ln(z³)) / (16z) dz
문제 8.1.50
Centroid: Find the centroid of the region bounded by the x-axis, the curve y = csc x, and the lines x = π/6, x = 5π/6.
문제 8.6.2
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ dx / (x √(x + 4))
문제 8.6.56
Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.
∫ (from 0 to 1/√3) dt / (t² + 1)^(7/2)
문제 8.6.24
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ sin(t / 3) sin(t / 6) dt
문제 8.8.52
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 4 to ∞ of (dx / (√x - 1))
문제 8.4.63
Find the average value of f(x) = (√(x + 1)) / √x on the interval [1, 3].
문제 8.8.56
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))
문제 8.2.46
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ √x e√x dx
문제 8.8.66
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)
문제 8.1.44
The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (tan θ + 3 / sin θ) dθ
문제 8.4.52
For Exercises 49–52, complete the square before using an appropriate trigonometric substitution.
∫ √(x² + 2x + 2) / (x² + 2x + 1) dx
문제 8.2.56
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ (sin⁻¹ x)² / √(1 - x²) dx
문제 8.7.35
[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.
Find, to two decimal places, the areas of the surfaces generated by revolving the curves in Exercises 35 and 36 about the x-axis.
y = sin x, 0 ≤ x ≤ π
문제 8.2.78
Use the formula ∫ f⁻¹(x) dx = x f⁻¹(x) - ∫ f(y) dy, y = f⁻¹(x)
To evaluate the integrals in Exercises 77-80. Express your answers in terms of x.
∫ arctan x dx
문제 8.6.26
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ cos(θ / 2) cos(7θ) dθ
문제 8.3.38
Evaluate the integrals in Exercises 33–52.
∫ sec⁴(x) tan²(x) dx
문제 8.4.56
Solve the initial value problems in Exercises 53–56 for y as a function of x.
(x² + 1)² (dy/dx) = √(x² + 1), where y(0) = 1
문제 8.7.32
The length of one arch of the curve y = sin x is given by
L = ∫(from 0 to π) √(1 + cos²(x)) dx.
Estimate L by Simpson's Rule with n = 8.
문제 8.3.50
Evaluate the integrals in Exercises 33–52.
∫ 8 cot⁴(t) dt
문제 8.2.4
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ x² sin(x) dx
문제 8.6.48
Use reduction formulas to evaluate the integrals in Exercises 41–50.
∫ 3 sec^4(3x) dx
문제 8.2.38
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ x⁵ e³ˣ dx
문제 8.3.76
Volume: Find the volume of the solid formed by revolving the region bounded by the graphs of y = sin x + sec x, y = 0, x = 0, and x = π/3 about the x-axis.
문제 8.8.2
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₁^∞ dx / x^1.001