11. Integrals of Inverse, Exponential, & Logarithmic Functions

37–56. Integrals Evaluate each integral.

∫ sech² w tanh w dw

37–56. Integrals Evaluate each integral.

∫ (cosh z) / (sinh² z) dz

57–58. Two ways

Evaluate the following integrals two ways.

a. Simplify the integrand first and then integrate.

b. Change variables (let u = ln x), integrate, and then simplify your answer. Verify that both methods give the same answer.

∫ (sinh (ln x)) / x dx

Area of region Find the area of the region bounded by y = sech x, x = 1, and the unit circle (see figure).

Solid of revolution Compute the volume of the solid of revolution that results when the region in Exercise 85 is revolved about the x-axis.

2–9. Integrals Evaluate the following integrals.

∫₀¹ (x² / (9 − x⁶)) dx

2–9. Integrals Evaluate the following integrals.

∫ dx / √(x² − 9),x > 3

Visual approximation

a. Use a graphing utility to sketch the graph of y = coth x and then explain why ∫₅¹⁰ coth x dx ≈ 5.

61–62. Points of intersection and area

b. Compute the area of the region described.

f(x) = sinh x, g(x) = tanh x; the region bounded by the graphs of f, g, and x = ln 3

Inverse identity Show that cosh⁻¹(cosh x) = |x| by using the formula cosh⁻¹ t = ln (t + √(t² – 1)) and considering the cases x ≥ 0 and x < 0.

Catenary arch The portion of the curve y =17/15 - cosh x that lies above the x-axis forms a catenary arch. Find the average height of the arch above the x-axis.

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 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.

∫ (dt / t√(3 + t²)

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 / ((x - 2)√(x² - 4x + 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.

∫ (√x / (1 + x³)) dx

Hint: Let u = x^(3/2).