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.

∫₋₁¹ (√(1 + x²) sin x) dx

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

∫ dx / (x √(x + 4))

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

∫₀^∞ dx / [(x + 1)(x² + 1)]

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 2 to ∞ of ((1 / ln x) dx)

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

∫₀⁴ dx / √(4 − x)

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

∫ dx / (x² √(4x - 9))

Evaluate the integrals in Exercises 33–52.

∫ eˣ sec³(eˣ) dx

In Exercises 27–40, use a substitution to change the integral into one you can find in the table. Then evaluate the integral.

∫ cos^(-1)(√x) / √x dx

Arc length: Find the length of the curve y = ln(sec x), 0 ≤ x ≤ π/4.

Evaluate the integrals in Exercises 33–52.

∫ cot³(t) csc⁴(t) dt

Evaluate the integrals in Exercises 33–52.

∫ cot⁶(2x) dx

Evaluate the integrals in Exercises 23–32.

∫₀^π √(1 - cos(2x)) 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.

∫ 3 sinh(x/2 + ln 5) dx

[Technology Exercise] 75. Find, to two decimal places, the x-coordinate of the centroid of the region in the first quadrant bounded by the x-axis, the curve y = arctan(x), and the line x = √3.

Use integration by parts to obtain the formula ∫ √(1 - x²) dx = (1/2) x √(1 - x²) + (1/2) ∫ 1 / √(1 - x²) 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 0 to ∞ of (dθ / (1 + e^θ))

Evaluate the integrals in Exercises 1–14.

∫ √(1 - 9t²) dt

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

z / (z³ - z² - 6z)

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

∫₋∞² (2 dx) / (x² + 4)

Evaluate the integrals in Exercises 1–22.

∫₀^π 8cos⁴(2πx) dx

Use any method to evaluate the integrals in Exercises 55–66.

∫ 2^x / (2²x + 2^x - 2) 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.

∫ (dθ / cos θ - 1)

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

∫ √x e√x dx

In Exercises 69–80, determine whether the improper integral converges or diverges. If it converges, evaluate the integral.

∫₋₂¹ (1 / x⁴) 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.

∫ (2^(√y) dy) / 2√y

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)

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.

∫ (x² / (x² + 1)) dx

Evaluate the integrals in Exercises 39–54.

∫ (e⁴t + 2e²t - e^t) / (e²t + 1) dt

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 - r²)^(5/2) / r⁸ dr

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.