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

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)

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

Expand the quotients in Exercises 1–8 by partial fractions.

(2x + 2) / (x² - 2x + 1)

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.

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

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.

Use the table of integrals at the back of the text to evaluate the integrals in Exercises 1–26.

∫ dx / (x √(x + 4))

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)

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

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))

Find the average value of f(x) = (√(x + 1)) / √x on the interval [1, 3].

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.

∫ (csc t sin 3t dt)

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.

∫ √x e√x dx

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)

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

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 / cos θ - 1)

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.

∫ (sin⁻¹ x)² / √(1 - x²) dx

Evaluate the integrals in Exercises 1–14.

∫ (2 dx) / √(1 - 4x²) from 0 to 1/(2√2)

[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

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.

∫ (1 - x²)^(1/2) / x⁴ dx

Evaluate the integrals in Exercises 33–52.

∫ sec⁴(x) tan²(x) dx

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

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.

Evaluate the integrals in Exercises 33–52.

∫ 8 cot⁴(t) dt

Evaluate the integrals in Exercises 1–24 using integration by parts.

∫ x² sin(x) dx

Use reduction formulas to evaluate the integrals in Exercises 41–50.

∫ 3 sec^4(3x) dx

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.

∫ x⁵ e³ˣ dx

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.

The integrals in Exercises 1–34 converge. Evaluate the integrals without using tables.

∫₁^∞ dx / x^1.001