11. Integrals of Inverse, Exponential, & Logarithmic Functions

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

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

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

76. Apparent discrepancy

Three different computer algebra systems give the following results:

∫ (dx / (x√(x⁴ − 1))) = ½ cos⁻¹(√(x⁻⁴)) = ½ cos⁻¹(x⁻²) = ½ tan⁻¹(√(x⁴ − 1)).

Explain how all three can be correct.

Evaluate the integrals in Exercises 53–76.

57. ∫dx/(x√(25x²-2))

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)))

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)

Find the limits in Exercises 1–6.

1. lim(b→1⁻) ∫(from 0 to b) dx/√(1-x²)