뒤로
문제 8.8.36
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 1 of (dθ / (θ² - 2θ))
문제 8.4.48
In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.
∫ √(x - 2) / √(x - 1) dx
문제 8.8.32
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀² dx / √|x − 1|
문제 8.5.10
In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ dx / (x² + 2x)
문제 8.3.70
Use any method to evaluate the integrals in Exercises 65–70.
∫ x cos³(x) dx
문제 8.3.14
Evaluate the integrals in Exercises 1–22.
∫₀^(π/2) sin²(x) dx
문제 8.8.4
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀⁴ dx / √(4 − x)
문제 8.5.22
In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (3t² + t + 4) / (t³ + t) dt from 1 to √3
문제 8.6.10
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ √(x - x²) / x dx
문제 8.4.58
Area: Find the area enclosed by the ellipse x²/a² + y²/b² = 1.
문제 8.3.4
Evaluate the integrals in Exercises 1–22.
∫ sin⁴(2x) cos(2x) dx
문제 8.6.6
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ x (7x + 5)^(3/2) dx
문제 8.8.86
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.
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 8.3.18
Evaluate the integrals in Exercises 1–22.
∫₀^π 8cos⁴(2πx) dx
문제 8.2.68
In Exercises 67–73, use integration by parts to establish the reduction formula.
∫ x^n sin(x) dx = -x^n cos(x) + n ∫ x^(n-1) cos(x) dx
문제 8.3.60
Exercises 59–64 require the use of various trigonometric identities before you evaluate the integrals.
∫ cos²(2θ) sin(θ) dθ
문제 8.4.63
Find the average value of f(x) = (√(x + 1)) / √x on the interval [1, 3].
문제 8.3.64
Exercises 59–64 require the use of various trigonometric identities before you evaluate the integrals.
∫ sin(θ) sin(2θ) sin(3θ) dθ
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 8.3.42
Evaluate the integrals in Exercises 33–52.
∫ tan⁴(x) sec³(x) 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 8.2.16
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ p⁴ e^(-p) dp
문제 8.4.61
Evaluate ∫ x³ √(1 - x²) dx using:
c. A trigonometric substitution.
문제 8.4.8
Evaluate the integrals in Exercises 1–14.
∫ √(1 - 9t²) dt
문제 8.3.74
Area: Find the area between the x-axis and the curve y = √(1 + cos 4x), for 0 ≤ x ≤ π.
문제 8.1.12
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.
∫₋₁³ (4x² - 7) / (2x + 3) dx
문제 8.6.16
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ e^(-3t) sin(4t) dt
문제 8.4.36
Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.
∫ (x dx) / (25 + 4x²)
문제 8.2.36
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ (ln x)³/x dx