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Problem 8.3.74
Area: Find the area between the x-axis and the curve y = √(1 + cos 4x), for 0 ≤ x ≤ π.
Problem 8.3.54
Evaluate the integrals in Exercises 53–58.
∫ sin(2x) cos(3x) dx
Problem 8.8.22
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀^∞ 2e^(−θ) sinθ dθ
Problem 8.8.18
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₁^∞ dx / [x√(x² − 1)]
Problem 8.3.34
Evaluate the integrals in Exercises 33–52.
∫ sec(x) tan²(x) dx
Problem 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
Problem 8.4.42
In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.
∫ dy / (y√(1 + (ln y)²)) from 1 to e
Problem 8.3.73
Volume: Find the volume generated by revolving one arch of the curve y = sin x about the x-axis.
Problem 8.1.26
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.
∫ (6 dy / √y(1 + y))
Problem 8.6.18
Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.
∫ x arctan(x) dx
Problem 8.5.2
Expand the quotients in Exercises 1–8 by partial fractions.
(5x - 7) / (x² - 3x + 2)
Problem 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
Problem 8.4.8
Evaluate the integrals in Exercises 1–14.
∫ √(1 - 9t²) dt
Problem 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
Problem 8.4.61
Evaluate ∫ x³ √(1 - x²) dx using:
c. A trigonometric substitution.
Problem 8.2.18
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫ (r² + r + 1) e^r dr
Problem 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
Problem 8.2.59
Finding volume: Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve y = e^x, and the line x = ln(2) about the line x = ln(2).
Problem 8.7.37
[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
arcsin(0.6) = ∫ (from 0 to 0.6) dx / √(1 - x²).
For reference, arcsin(0.6) = 0.64350 to five decimal places.
Problem 8.3.58
Evaluate the integrals in Exercises 53–58.
∫ from -π/2 to π/2 of cos(x) cos(7x) dx
Problem 8.3.56
Evaluate the integrals in Exercises 53–58.
∫ from 0 to π/2 of sin(x) cos(x) dx
Problem 8.1.18
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^(√y) dy) / 2√y
Problem 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
Problem 8.6.54
Evaluate the integrals in Exercises 51–56 by making a substitution (possibly trigonometric) and then applying a reduction formula.
∫ (from 0 to √3/2) dy / (1 - y²)^(5/2)
Problem 8.8.30
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₂⁴ dt / [t√(t² − 4)]
Problem 8.5.8
Expand the quotients in Exercises 1–8 by partial fractions.
(t⁴ + 9) / (t⁴ + 9t²)
Problem 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))
Problem 8.5.4
Expand the quotients in Exercises 1–8 by partial fractions.
(2x + 2) / (x² - 2x + 1)
Problem 8.1.4
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.
∫ (1 / (cos² x tan x)) dx from π/3 to π/4
Problem 8.1.22
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.
∫ (x + 2√(x - 1)) / (2x√(x - 1)) dx