Determine whether or not each of the following functions is integrable over the proposed interval (see Example \ref{exam 8.6.3}). Give a reason for your answer.

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• [math]\frac{x^2+x-2}{x-1}[/math], over [math][0,1][/math]
• [math]\frac{x^2+x-2}{x-1}[/math], over [math][0,2][/math]
• [math]\frac{x^2+x+2}{x-1}[/math], over [math][0,2][/math].

Is each of the following integrals defined? (See Example \ref{exam 8.6.3}.) Give a reason for your answer.

• [math]\int_0^1 \frac{\sin x}x dx[/math]
• [math]\int_0^{\frac12} \frac{\tan 2x}x dx[/math]
• [math]\int_0^1 \frac1x dx[/math]
• [math]\int_0^{\frac1e} \frac1{\ln x}dx[/math]
• [math]\int_0^e \ln x \; dx[/math].

Draw the graph of [math]f[/math], and evaluate [math]\int_a^b f(x) \; dx[/math] in each of the following examples.

• [math]\trilemma{1 & \mbox{if [/math]-\infty < x \leq 0[math]},} {5 & \mbox{if [/math]0 < x < 2[math]},} {3 & \mbox{if [/math]2 \leq x < \infty[math]},}[/math] and [math][a,b] = [-3,3][/math].
• [math]f(x) = \dilemma{x^2&\mbox{if [/math]-\infty < x < 0[math]},} {2-x^2&\mbox{if [/math]0\leq x < \infty[math]},}[/math] and [math][a,b] = [-2,2][/math].
• [math]f(x) = n[/math] \quad if [math]n \leq x \lt n+1[/math] where [math]n[/math] is any integer, and [math][a,b] = [0,5][/math].

Prove that if a function [math]f[/math] is bounded on an open interval [math](a,b)[/math] and, if [math]f(a)[/math] and [math]f(b)[/math] are any two real number, then [math]f[/math] is also bounded on the closed interval [math][a,b][/math].

Compute

• [math]\lim_{n\goesto0+} \int_t^1 \frac1x dx[/math]
• [math]\lim_{t\goesto1-} \int_0^t \tan \frac{\pi}2 x \; dx[/math].

How does the result give insight into the fact that neither integrand is integrable over the interval [math][0,1][/math]?

A function [math]f[/math] is said to be piecewise continuous on an interval [math][a,b][/math] if it is continuous at all but possibly a finite number of points of the interval, and if, for every point [math]c[/math] of discontinuity in the interval, there exist number [math]k[/math] and [math]l[/math] such that

Using Theorems \ref{thm 8.6.2} and \ref{thm 8.6.3}, prove that if [math]f[/math] is piecewise continuous on [math][a,b][/math], then it is integrable over [math][a,b][/math].