11. Integrals of Inverse, Exponential, & Logarithmic Functions

2–9. Integrals Evaluate the following integrals.

∫ (x + 4) / (x² + 8x + 25) dx

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume x > 0 and y > 0.

e. The area under the curve y = 1/x and the x-axis on the interval [1, e] is 1.

49–63. {Use of Tech} Integrating with a CAS Use a computer algebra system to evaluate the following integrals. Find both an exact result and an approximate result for each definite integral. Assume a is a positive real number.

61. ∫₀¹ (ln x) ln(1 + x) dx

Evaluate the integrals in Exercises 39–56.

41. ∫2y dy/(y²-25)

Evaluate the integrals in Exercises 39–56.

55. ∫dx/(2√x + 2x)

Evaluate the integrals in Exercises 97–110.

109. ∫ (dx / (x log₁₀x))

Evaluate the integrals in Exercises 31–78.

45. ∫(ln x)^(-3)/x dx

Evaluate the integrals in Exercises 31–78.

61. ∫(from 1 to 3)(ln(v+1))²/(v+1) dv

Evaluate the integrals in Exercises 31–78.

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

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 - √x)

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)

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 97–110.

101. ∫ (log₁₀x / x) dx

Evaluate the integrals in Exercises 97–110.

103. ∫₁⁴ (ln 2 · log₂x / x) dx

Evaluate the integrals in Exercises 97–110.

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