11. Integrals of Inverse, Exponential, & Logarithmic Functions

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).

Evaluate the integrals in Exercises 1–14.

∫ (2 dx) / (x³ √(x² - 1)), where x > 1

Use any method to evaluate the integrals in Exercises 15–38. Most will require trigonometric substitutions, but some can be evaluated by other methods.

∫ (x² dx) / (4 + x²)

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 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 1–14.

∫ dx / (8 + 2x²) from 0 to 2

Evaluate the integrals in Exercises 1–14.

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

Evaluate the integrals in Exercises 53–76.

53. ∫dx/√(9-x²)

Evaluate the integrals in Exercises 53–76.

55. ∫dx/(17+x²)

Evaluate the integrals in Exercises 53–76.

57. ∫dx/(x√(25x²-2))

Evaluate the integrals in Exercises 53–76.

59. ∫(from 0 to 1)4ds/√(4-s²)