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)

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.

∫ (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)