11. Integrals of Inverse, Exponential, & Logarithmic Functions

Zero net area Consider the function f(x) = (1 − x)/x

a. Are there numbers 0 < a < 1 such that ∫₁₋ₐ¹⁺ᵃ f(x) dx = 0?

Evaluate the integrals in Exercises 87–96.

95. ∫₂⁴ x^(2x) (1 + ln x) dx

Evaluate the integrals in Exercises 97–110.

107. ∫₀⁹ (2 log₁₀(x + 1) / (x + 1)) dx

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

∫ (x dx) / (25 + 4x²)

Evaluate the integrals in Exercises 31–78.

52. ∫(from 1 to 32)(1/5x) dx

Use any method to evaluate the integrals in Exercises 55–66.

∫ 2 / (x(ln x - 2)³) dx

Evaluate the integrals in Exercises 1–6.

∫ dt / (t - √(1 - 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.

∫ (x + 2√(x - 1)) / (2x√(x - 1)) dx

Find the indefinite integral.

${\int }_{}^{}\frac{3-y^2}{2y}dy$

Evaluate the definite integral.

${\int }_1^6\frac{dt}{2+5t}$

Evaluate the integrals in Exercises 97–110.

105. ∫₀² (log₂(x + 2) / (x + 2)) dx

Evaluate the integrals in Exercises 31–78.

64. ∫(from 1 to e)(8ln3 log_3(θ))/θ dθ

37–56. Integrals Evaluate each integral.

∫ dx/(8 – x²), x > 2√2

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

∫ √(1 - (ln x)²) / (x ln x) dx

Evaluate the integrals in Exercises 31–78.

41. ∫(from 0 to 4)2t/(t² - 25)dt