11. Integrals of Inverse, Exponential, & Logarithmic Functions

Evaluate the integrals in Exercises 31–78.

69. ∫dy/(y√(4y²-1))

Verify the integration formulas in Exercises 37–40.

37. b. ∫sech(x)dx = sin⁻¹(tanh x) + C

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

Evaluating integrals Evaluate the following integrals.

∫√₂/₅^²/⁵ d𝓍/𝓍√(25𝓍² ―1)

Verify the integration formulas in Exercises 37–40.

37. a. ∫sech(x)dx = tan⁻¹(sinh x) + C

Evaluate the integrals in Exercises 1–14.

∫ dx / (8 + 2x²) from 0 to 2

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.

In Exercises 39–48, use an appropriate substitution and then a trigonometric substitution to evaluate the integrals.

∫ √(x) / (1 - x³) dx (Hint: Let u = x³/2)

Evaluate the integrals in Exercises 53–76.

65. ∫3dr/√(1-4(r-1)²)

Evaluate the definite integral in terms of an inverse trig function.

$\(\int\)_0^{\(\frac{\sqrt3}{3}\)}\(\frac{dx}{\sqrt{4-9x^2}\)}$

Evaluate the integrals in Exercises 67–74 in terms of

a. inverse hyperbolic functions.

69. ∫(from 5/4 to 2)dx/(1-x²)

37–56. Integrals Evaluate each integral.

∫ sech² w tanh w dw

Verify the integration formulas in Exercises 37–40.

39. ∫x coth⁻¹(x)dx = ((x²-1)/2)coth⁻¹(x) + x/2 + C

37–56. Integrals Evaluate each integral.

∫ tanh²x dx (Hint: Use an identity.)

37–56. Integrals Evaluate each integral.

∫ sinh x / (1 + cosh x) dx