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Problem 8.4.26
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 √(x² - 4) dx
Problem 8.5.12
In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (2x + 1) / (x² - 7x + 12) dx
Problem 8.8.16
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀² (s + 1) / √(4 − s²) ds
Problem 8.3.6
Evaluate the integrals in Exercises 1–22.
∫ cos³(4x) dx
Problem 8.8.32
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀² dx / √|x − 1|
Problem 8.8.34
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₀^∞ dx / [(x + 1)(x² + 1)]
Problem 8.4.10
Evaluate the integrals in Exercises 1–14.
∫ 5 dx / √(25x² - 9), where x > 3/5
Problem 8.2.40
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ x² sin(x³) dx
Problem 8.2.8
Evaluate the integrals in Exercises 1–24 using integration by parts.
∫x e^(3x) dx
Problem 8.3.48
Evaluate the integrals in Exercises 33–52.
∫ cot⁶(2x) dx
Problem 8.2.54
Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.
∫ (xe^x) / (x + 1)² dx
Problem 8.2.76
Use integration by parts to obtain the formula ∫ √(1 - x²) dx = (1/2) x √(1 - x²) + (1/2) ∫ 1 / √(1 - x²) dx.
Problem 8.3.40
Evaluate the integrals in Exercises 33–52.
∫ eˣ sec³(eˣ) dx
Problem 8.8.10
The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.
∫₋∞² (2 dx) / (x² + 4)
Problem 8.3.46
Evaluate the integrals in Exercises 33–52.
∫ from -π/4 to π/4 of 6 tan⁴(x) dx
Problem 8.4.22
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 / (x² √(x² + 1))
Problem 8.9.40
Annual rainfall The annual rainfall in inches for San Francisco, California, is approximately a normal random variable with mean 20.11 in. and standard deviation 4.7 in. What is the probability that next year’s rainfall will exceed 17 in.?
Problem 8.5.52
Evaluate the integrals in Exercises 39–54.
∫ 1 / (cos θ + sin 2θ) dθ
Problem 8.3.70
Use any method to evaluate the integrals in Exercises 65–70.
∫ x cos³(x) dx
Problem 8.1.40
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 / (1 + x³)) dx
Hint: Let u = x^(3/2).
Problem 8.1.20
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.
∫ (dt / t√(3 + t²)
Problem 8.5.10
In Exercises 9–16, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ dx / (x² + 2x)
Problem 8.3.14
Evaluate the integrals in Exercises 1–22.
∫₀^(π/2) sin²(x) dx
Problem 8.5.38
In Exercises 33–38, perform long division on the integrand, write the proper fraction as a sum of partial fractions, and then evaluate the integral.
∫ 2y⁴ / (y³ - y² + y - 1) dy
Problem 8.3.8
Evaluate the integrals in Exercises 1–22.
∫₀^π sin⁵(x/2) dx
Problem 8.3.60
Exercises 59–64 require the use of various trigonometric identities before you evaluate the integrals.
∫ cos²(2θ) sin(θ) dθ
Problem 8.5.30
In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.
∫ (x² + x) / (x⁴ - 3x² - 4) dx
Problem 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)
Problem 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
Problem 8.2.30
Evaluate the integrals in Exercises 25–30 by using a substitution prior to integration by parts.
∫ z(ln z)² dz