11. Integrals of Inverse, Exponential, & Logarithmic Functions

Evaluate the integrals in Exercises 39–56.

55. ∫dx/(2√x + 2x)

Evaluate the integrals in Exercises 67–74 in terms of

b. natural logarithms.

73. ∫(from 0 to π)cos(x)dx/√(1+sin²x)

Find the area under the graph of $f\left(x\right)=\frac{e^{-x}}{1+e^{-x}}$ from $x=0$ to $x=4$.

Evaluate the integrals in Exercises 31–78.

45. ∫(ln x)^(-3)/x dx

Evaluate the integrals in Exercises 53–76.

61. ∫(from 0 to 2)dt/√(8+2t²)

Evaluate the integrals in Exercises 97–110.

109. ∫ (dx / (x log₁₀x))

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)

Solve the initial value problem given by $\frac{dy}{dx}=\frac{2}{x}+3,y\left(1\right)=4$.

7-56. Trigonometric substitutions Evaluate the following integrals using trigonometric substitution.

19. ∫ 1/√(x² - 81) dx, x > 9

Evaluate the integrals in Exercises 39–56.

41. ∫2y dy/(y²-25)

Evaluate the integrals in Exercises 53–76.

69. ∫dx/((2x-1)√((2x-1)²-4))

Evaluate the integrals in Exercises 31–56. Some integrals do not require integration by parts.

∫ (ln x)³/x dx

Evaluate the integrals in Exercises 97–110.

103. ∫₁⁴ (ln 2 · log₂x / x) 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.

∫ 3 sinh(x/2 + ln 5) dx

Evaluate the integrals in Exercises 67–74 in terms of

b. natural logarithms.

71. ∫(from 1/5 to 3/13)dx/(x√(1-16x²))