Area: Find the area enclosed by the ellipse x²/a² + y²/b² = 1.

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 25–30 by using a substitution prior to integration by parts.

∫ from 0 to 1 x√(1 - x) dx

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

∫ x² sin(x³) 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.

∫₋₁¹ (√(1 + x²) sin 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.

∫ (2^(√y) dy) / 2√y

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 1 of (dθ / (θ² - 2θ))

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

∫₋∞^∞ (x dx) / (x² + 4)^(3/2)

Evaluate the integrals in Exercises 33–52.

∫ cot⁶(2x) dx

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

z / (z³ - z² - 6z)

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

∫ (x² - 2x + 1) e^(2x) 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 2 to ∞ of ((1 / ln x) dx)

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

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

∫ dx / (x² √(4x - 9))

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 -∞ to ∞ of ((dx) / (e^x + e^(-x)))

Solve the initial value problems in Exercises 67–70 for x as a function of t.

(t + 1) (dx/dt) = x² + 1 (for t > -1), x(0) = 0

Exercises 83–86 are about the infinite region in the first quadrant between the curve y = e^(-x) and the x-axis.

83. Find the area of the region.

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

∫₋∞² (2 dx) / (x² + 4)

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

∫ cos(θ / 2) cos(7θ) dθ

Evaluate the integrals in Exercises 1–14.

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

[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.

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

∫₋₂¹ (1 / x⁴) dx

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

∫ √x e√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.

∫ (dx / ((2x + 1)√(4x + 4x²)))

In Exercises 21–32, express the integrand as a sum of partial fractions and evaluate the integrals.

∫ (8x² + 8x + 2) / (4x² + 1)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.

∫ (x² / (x² + 1)) dx

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

∫₀^∞ 2e^(−θ) sinθ dθ

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

∫ (x + 2) / (x³ - 2x² - 3x) dx

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.