Find the derivative with respect to [math]x[/math] of each of the following functions.

• [a [math]\ln x^2[/math]
• [math]\ln(7x + 2)[/math]
• [math]\ln \sqrt{(x-3)(x+4)}[/math]
• [math]\ln(x^2 - 9x + 3)[/math]
• [math]\ln frac2x[/math]
• [math]\ln \sqrt[7]{7x^3}[/math]
• [math]\int_1^{x^2+3} \frac{dt}t[/math]
• [math](\ln x)^3[/math]
• [math]\ln(\ln x)[/math]
• [math]\ln x\sqrt{x-1}[/math]
• [math]\ln \frac{x-3}{x+1}[/math]
• [math]\ln \frac{x^2-2x+4}{x^2+1}[/math]
• [math]\ln \frac{x}{2-x^2}[/math].

Show that [math]|x^2 + 2x + 3| = x^2 + 2x + 3[/math] and hence that [math]\ln(x^2 + 2x + 3)[/math] is defined for all [math]x[/math].

If [math]\ln 2 = p[/math], [math]\ln 3 = q[/math], and [math]\ln 5 = r[/math], write each of the following as a function of [math]p[/math], [math]q[/math], and [math]r[/math].

• [math]\ln 10[/math]
• [math]\ln 0.25[/math]
• [math]\ln 6000[/math]
• [math]\ln 0.625[/math]
• [math]\ln 0.03[/math]
• [math]\ln 1728[/math].

Integrate each of the following.

• [math]\int \frac{2\;dx}x[/math]
• [math]\int \frac{x\;dx}{x^2+1}[/math]
• [math]\int \frac{(x-3)\;dx}{x^2-6x+2}[/math]
• [math]\int x\sqrt{x^2+3}\;dx[/math]
• [math]\int \frac{13x^2}{x^3-6}\;dx[/math]
• [math]\int \left(\frac2{x+1} + \frac3{2x-1} - \frac4{3x+5}\right)dx[/math]
• [math]\int \frac{x\;dx}{x-1}[/math]
• [math]\int \frac{2\;dx}{(2x-1)^2}[/math]
• [math]\int \frac1x \ln x\;dx[/math].

Sketch the graph of each of the following functions. Label all the extreme points and points of inflection, and give the values of [math]x[/math] at which these occur. Classify each extreme point as a local maximum or minimum.

• [math]f(x) = \ln(x+4)[/math]
• [math]f(x) = \ln x^2[/math]
• [math]f(x) = \ln(1+x^2)[/math]
• [math]f(x) = x^2 - 4 \ln x^2[/math]
• [math]f(x) = x^2 + 4 \ln x^2[/math].

Compute the following definite integrals.

• [math]\int_0^1 \frac{dt}{t+1}[/math]
• [math]\int_1^3 \frac{dx}{5x-3}[/math]
• [math]\int_0^1 \frac{s-2}{s^2-4s+4} \; ds[/math]
• [math]\int_2^x \frac{t\;dt}{t^2-3}[/math]
• [math]\int_1^{x^2} \frac{dt}{t^2}[/math]
• [math]\int_{-5}^{-1} \frac{dx}x[/math].

In each of the following examples, find the area of the region bounded by the graph of [math]y = f(x)[/math], the [math]x[/math]-axis, and the two vertical lines whose equations are given.

• [math]f(x) = \frac1x[/math], [math]x=3[/math] and [math]x=7[/math].
• [math]f(x) = \frac{\ln x}{x}[/math], [math]x=1[/math] and [math]x=4[/math].
• [math]f(x) = \frac1{x^2}[/math], [math]x=9[/math] and [math]x=11[/math].

If [math]F(x) = \int_2^x \frac{t\;dt}{t^2-3}[/math], what is the domain of the function [math]F[/math]?

• For what values of [math]x[/math] is [math]\int_{-2}^x \frac{dt}{t-1}[/math] defined?
• What, if anything, is wrong with the computation
  [[math]] \int_{-2}^2 \left.\frac{dx}{x-1} = \ln|x-1| \; \right|_{-2}^2 = \ln 1 - \ln 3 = -\ln 3 ? [[/math]]

Use the appropriate form of L'H\^opital's Rule to evaluate each of the following limits.

• [math]\lim_{x\goesto\infty} \frac{\ln x}x[/math]
• [math]\lim_{x\goesto\infty} \frac{\ln x}{x^n}[/math], [math]n[/math] an arbitrary positive integer.
• [math]\lim_{x\goesto0+} x \ln x[/math]
• [math]\lim_{x\goesto1} \frac{\ln x}{1-x}[/math]
• [math]\lim_{x\goesto0+} x^n \ln x[/math], [math]n[/math] an arbitrary positive integer.