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문제 8.2.30
Evaluate the integrals in Exercises 25–30 by using a substitution prior to integration by parts.
∫ z(ln z)² dz
문제 8.7.36
[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 = x²/4, 0 ≤ x ≤ 2
문제 8.1.6
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.
∫ dx / (x - √x)
문제 8.6.52
Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.
∫ csc³(√θ) / √θ dθ
문제 8.8.54
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 ∞ of (dθ / (1 + e^θ))
문제 8.8.46
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 (dt / (t - sin t))
(Hint: t ≥ sin t for t ≥ 0)
문제 8.3.20
Evaluate the integrals in Exercises 1–22.
∫₀^π 8 sin⁴(y) cos²(y) dy
문제 8.3.48
Evaluate the integrals in Exercises 33–52.
∫ cot⁶(2x) dx
문제 8.5.58
Use any method to evaluate the integrals in Exercises 55–66.
∫ 2^x / (2²x + 2^x - 2) dx
문제 8.1.16
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.
∫ (dθ / √(2θ - θ²))
문제 8.3.46
Evaluate the integrals in Exercises 33–52.
∫ from -π/4 to π/4 of 6 tan⁴(x) dx
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 8.4.48
In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.
∫ √(x - 2) / √(x - 1) dx
문제 8.5.20
In Exercises 17–20, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (x² dx) / ((x - 1)(x² + 2x + 1))
문제 8.3.70
Use any method to evaluate the integrals in Exercises 65–70.
∫ x cos³(x) dx
문제 8.8.34
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀^∞ dx / [(x + 1)(x² + 1)]
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 8.6.6
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 8.5.52
Evaluate the integrals in Exercises 39–54.
∫ 1 / (cos θ + sin 2θ) dθ
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 8.9.18
Find the value of the constant c so that the given function is a probability density function for a random variable X over the specified interval.
f(x) = (1/x) over [c, c + 1]
문제 8.2.70
In Exercises 67–73, use integration by parts to establish the reduction formula.
∫ (ln x)^n dx = x (ln x)^n - n ∫ (ln x)^(n-1) dx
문제 8.3.78
Average Value: Find the average value of the function f(x) = 1 / (1 - sin θ) on the interval [0, π/6].
문제 8.3.14
Evaluate the integrals in Exercises 1–22.
∫₀^(π/2) sin²(x) dx
문제 8.7.38
[Technology Exercise] When solving Exercises 33-40, you may need to use a calculator or a computer.
Use numerical integration to estimate the value of
π = 4 ∫ (from 0 to 1) [ 1 / (1 + x²) ] dx.
문제 8.3.10
Evaluate the integrals in Exercises 1–22.
∫₀^(π/6) 3cos⁵(3x) dx
문제 8.8.6
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₋₈¹ dx / x^(1/3)
문제 8.2.40
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ x² sin(x³) dx
문제 8.5.56
Use any method to evaluate the integrals in Exercises 55–66.
∫ (x + 2) / (x³ - 2x² - 3x) dx