Classify each of the following integrals as proper, improper and convergent, or improper and divergent. Evaluate any which are convergent if an indefinite integral can be found.

• [a [math]\int_0^2 \frac3{x^{\frac23}} dx[/math]
• [math]\int_1^2 \frac1{\sqrt{2-x}}dx[/math]
• [math]\int_2^3 \frac1{(x-2)^2} dx[/math]
• [math]\int_0^1 \frac1{x^2+x+1} dx[/math]
• [math]\int_0^{\frac{\pi}2} \tan x \; dx[/math]
• [math]\int_0^1 \frac{\sin x}x dx[/math].

Classify each of the following integrals and evaluate any which are not divergent.

• [math]\int_2^{\infty} \frac1{x^3} dx[/math]
• [math]\int_0^2 \frac1{x^3} dx[/math]
• [math]\int_{-1}^{\infty} (x^2-x+1) \; dx[/math]
• [math]\int_0^{\infty} xe^{-x^2} dx[/math]
• [math]\int_1^{\infty} \sin x \; dx[/math]
• [math]\int_{-\infty}^1 e^xdx[/math]
• [math]\int_0^{\infty} \frac1{(x+2)^2} dx[/math]
• [math]\int_0^1 x \ln x \; dx[/math].

Show that the integral [math]\int_0^1 \frac1{x^s} dx[/math] is

• proper if [math]-\infty \lt s \leq 0[/math].
• improper and convergent if [math]0 \lt s \lt 1[/math].
• improper and divergent if [math]1 \leq s \lt \infty[/math].

For what values of [math]s[/math] is the integral [math]\int_1^{\infty} \frac{dx}{x^s}[/math] convergent, and for what values is it divergent? Give reasons for your answers.

Classify each of the following integrals, and evaluate any which are not divergent if an indefinite integral can be found.

• [math]\int_{-1}^1 \frac1{x^{\frac23}} dx[/math]
• [math]\int_0^2 \frac1{(x-1)^{\frac13}} dx[/math]
• [math]\int_0^1 \frac{\tan x}x dx[/math]
• [math]\int_0^{\infty} \frac1{x^s} dx[/math]
• [math]\int_{-infty}^0 \frac{dx}{(x-2)^2}[/math]
• [math]\int_{-infty}^{infty} e^{-|x|} dx[/math]
• [math]\int_2^{\infty} \frac1{\sqrt{x-2}} dx[/math]
• [math]\int_0^2 \frac1{(x+1)(x-1)} dx[/math]
• [math]\int_1^{\infty} \frac{\ln x}x dx[/math]
• [math]\int_1^{\infty} \frac{\ln x}{x^2} dx[/math].

Prove Theorem \ref{thm 8.7.1}.

Using the Comparison Test for Integrals if necessary, classify each of the following integrals.

• [math]\int_{-\infty}^0 e^{-x^2} dx[/math]
• [math]\int_1^{\infty} \frac1{x^2} \sin x \; dx[/math]
• [math]\int_0^{\infty} e^{-x} \sin x \; dx[/math]
• [math]\int_3^{\infty} \frac1{\sqrt{(x-1)(x-2)}} dx[/math]
• [math]\int_0^1 x \sin \frac1x dx[/math]
• [math]\int_0^1 \frac1{\sqrt{(x-1)(x-2)}} dx[/math].

If [math]F(t) = \int_{-\infty}^t e^{-x^2} dx[/math], find [math]F^\prime(0)[/math] and [math]F^\prime(1)[/math].

• Show that the area of the region [math]P[/math] bounded by the [math]x[/math]-axis, the line [math]x=1[/math], and the curve [math]y=\frac1x[/math] is infinite.
• Show that the volume of the solid of revolution obtained by rotation the region [math]P[/math] about the [math]x[/math]-axis is finite.

Prove that if [math]f[/math] is bounded on [math](a,b][/math] and integrable over [math][t,b][/math] for every [math]t[/math] in [math](a,b][/math], then [math]f[/math] is integrable over [math][a,b][/math] and [math]\lim_{t\goesto a+} \int_t^b f = \int_a^b f[/math]. [Hint: The argument is essentially the same as that in the proof of Theorem \ref{thm 8.6.1}.]