11. Integrals of Inverse, Exponential, & Logarithmic Functions

Evaluate the integral.

${\int }_{\sqrt{2}}^2\frac{1}{\mid x\mid \sqrt{x^2-1}}dx$

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.

Visual approximation

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

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.

Evaluate the integrals in Exercises 53–76.

53. ∫dx/√(9-x²)

37–56. Integrals Evaluate each integral.

∫ (cosh z) / (sinh² z) dz

Find the indefinite integral.

${\int }_{}^{}\frac{arcsi{n}^{4}x}{\sqrt{1-x^2}}dx$

Using different substitutions

Show that the integral

∫((x² - 1)(x + 1))^(-2/3) dx

can be evaluated with any of the following substitutions.

f. u = arccos x

What is the value of the integral?

Evaluate the integrals in Exercises 53–76.

71. ∫(from -π/2 to π/2) 2cosθ dθ/(1+(sinθ)²)

60–69. Completing the square Evaluate the following integrals.

65. ∫[1/2 to (√2 + 3)/(2√2)] dx / (8x² - 8x + 11)

Evaluate the integrals in Exercises 77–90.

84. ∫(from 2 to 4)2dx/(x²-6x+10)

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).

Evaluate the integrals in Exercises 53–76.

55. ∫dx/(17+x²)

Evaluate the integrals in Exercises 91–102.

102. ∫(from -1/3 to 1/√3)(cos(arctan 3x))/(1+9x²) dx

37–56. Integrals Evaluate each integral.

∫ cosh 2x dx