11. Integrals of Inverse, Exponential, & Logarithmic Functions

29–62. Integrals Evaluate the following integrals. Include absolute values only when needed.

∫₁² (1 + ln x) x^x dx

Find the area under the graph of $f\left(t\right)=\frac{t^2{e}^t+t}{t^2}$ from $t=1$ to $t=3$.

Evaluate the integrals in Exercises 97–110.

101. ∫ (log₁₀x / x) dx

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

129. ∫ (x^(ln x) * ln x) / x dx

Evaluate the definite integral.

${\int }_0^{\frac{\pi }{3}}\frac{sin\theta }{1+\cos \theta }d\theta$

2–9. Integrals Evaluate the following integrals.

∫ (x + 4) / (x² + 8x + 25) dx

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

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.

63–68. Definite integrals Evaluate the following definite integrals. Use Theorem 7.7 to express your answer in terms of logarithms.

∫₁ᵉ^² dx/x√(ln²x + 1)

Find the indefinite integral.

${\int }_{}^{}\frac{1}{2x+5}dx$

Evaluate the definite integral.

${\int }_2^e\frac{3+x}{x}dx$

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.

∫ (2 ln(z³)) / (16z) dz

Evaluate the integrals in Exercises 31–78.

61. ∫(from 1 to 3)(ln(v+1))²/(v+1) dv

The region between the curve $y=\frac{2}{\sqrt{x}}$ and the $x$-axis from $x=1$ to $x=3$ is revolved about the $x$-axis to form a solid. Find the volume of this solid. 

Evaluate the integrals in Exercises 31–78.

37. ∫(from -1 to 1)dx/(3x-4)