11. Integrals of Inverse, Exponential, & Logarithmic Functions

Evaluate the integrals in Exercises 53–76.

75. ∫y dy/√(1-y^4)

Evaluate the integrals in Exercises 91–102.

96. ∫dy/((arcsin y)(1-y²))

Evaluate the integrals in Exercises 31–78.

75. ∫(from -2 to -1)2dv/(v²+4v+5)

Evaluate the integrals in Exercises 91–102.

93. ∫(arcsin x)²dx/√(1-x²)

Evaluate the integrals in Exercises 53–76.

59. ∫(from 0 to 1)4ds/√(4-s²)

Evaluate the integrals in Exercises 77–90.

90. ∫dx/((x-2)√(x²-4x+3))

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

∫ tan^(-1)(√y) dy

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)