Evaluate the integrals in Exercises 39–56.
55. ∫dx/(2√x + 2x)

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 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