Exercises: Improper Integrals

$\newenvironment {prompt}{}{} \newcommand {\ungraded }[0]{} \newcommand {\bigmath }[1]{$\displaystyle #1$} \newcommand {\choicebreak }[0]{} \newcommand {\type }[0]{} \newcommand {\notes }[0]{} \newcommand {\keywords }[0]{} \newcommand {\offline }[0]{} \newcommand {\comments }[0]{\begin {feedback}} \newcommand {\multiplechoice }[0]{\begin {multipleChoice}} \newcommand {\HyperFirstAtBeginDocument }[0]{\AtBeginDocument }$

Various exercises relating to improper integrals.

Evaluate the improper integral: $\displaystyle \int _0^\infty e^{5-2x}\ dx = \answer {(e^5)/2}.$

Evaluate the given improper integral: $\displaystyle \int _{-\infty }^\infty \frac {1}{x^2+9}\ dx = \answer {\pi /3}.$

Evaluate the integral: $\int _0^1 \frac {e^{\sqrt {x}}}{\sqrt {x}} \ dx = \answer {2(e-1)}.$ This integral is because of the behavior of the integrand near $x = 0$ .

Evaluate the given improper integral. $\displaystyle \int _{3}^\infty \frac {1}{x^2-4}\ dx = \answer {\frac {\ln 5}{4}}.$

Use the Direct Comparison Test or the Limit Comparison Test to determine whether the integral converges or diverges: $\displaystyle \int _{10}^\infty \frac {3}{\sqrt {3x^2+2x-5}} \ dx.$ Answer: the integral by comparison with the function $\displaystyle \frac {1}{x^{\answer {1}}}.$

Use the Direct Comparison Test or the Limit Comparison Test to determine whether the integral converges or diverges: $\displaystyle \int _{2}^\infty \frac {2}{\sqrt {7x^3-x}} \ dx.$ Answer: the integral by comparison with the function $\displaystyle \frac {1}{x^{\answer {3/2}}}$ (select the largest exponent for the denominator which makes the statement true).

Use the Direct Comparison Test or the Limit Comparison Test to determine whether the integral converges or diverges: $\displaystyle \int _{1}^\infty e^{-x} \ln x \ dx.$ Answer: the integral by direct comparison with the function

Use the Direct Comparison Test or the Limit Comparison Test to determine whether the integral converges or diverges: $\displaystyle \int _{1}^\infty e^{-x^2 + 3x + 1} \ dx.$ Answer: the integral by direct comparison with the function

Use the Direct Comparison Test or the Limit Comparison Test to determine whether the integral converges or diverges: $\displaystyle \int _{1}^\infty \frac {x}{x^2+\cos x} \ dx.$ Answer: the integral by comparison with the function

Use the Direct or Limit Comparison Test to determine whether the integral converges or diverges: $\int _0^{1/e} \frac {(\ln x)^2-1}{x^3 + x^2 + x} dx$ Answer: The integral by comparison with the function

Use the Direct or Limit Comparison Test to determine whether the integral converges or diverges: $\int _0^{1/e} \frac {(\ln x)^2-1}{x + \sqrt {x} + e^{-1/x}} dx$ Answer: The integral by direct comparison with the function

Sample Quiz Questions

Which of the following improper integrals is convergent? Show how you used comparison tests to justify your answer. $\mathrm {I}: \ \int _0^1\frac {\sqrt {{1}+{x^2}{e^{-x}}}}{{(\cos x)}{x^2}}~dx \qquad \mathrm {II}: \ \int _{2}^\infty \frac {{e^{x}}}{{x}{e^{x}}+{x^2}}~dx \qquad \mathrm {III}: \ \int _{2}^\infty \frac {{x^2}}{{x^4}+{1}}~dx$

Integral $\mathrm {I}$ is divergent by direct comparison to the function $\displaystyle \frac {{1}}{{x^2}}$ . Integral $\mathrm {II}$ is divergent by limit comparison to the function $\displaystyle \frac {{1}}{{x}}$ . Integral $\mathrm {III}$ is convergent by direct comparison to the function $\displaystyle \frac {{1}}{{x^2}}$ .

Which of the following improper integrals is convergent? Show how you used comparison tests to justify your answer. $\mathrm {I}: \ \int _0^1\frac {\sqrt {{e^{2x}}+{x^3}}}{{x}}~dx \qquad \mathrm {II}: \ \int _0^1\frac {{x^2}}{{x^2}{\sqrt {x}}+{x^3}}~dx \qquad \mathrm {III}: \ \int _{2}^\infty \frac {{x^2}{\ln x}}{-{x}+{x^4}}~dx$

Integral $\mathrm {I}$ is divergent by direct comparison to the function $\displaystyle \frac {{1}}{{x}}$ . Integral $\mathrm {II}$ is convergent by direct comparison to the function $\displaystyle \frac {{1}}{{\sqrt {x}}}$ . Integral $\mathrm {III}$ is convergent by limit comparison to the function $\displaystyle \frac {{\ln x}}{{x^2}}$ .

Sample Exam Questions

Only one of the following four improper integrals diverges. Choose that improper integral and justify why it diverges. (You need NOT justify why the other integrals converge.)

Press...	...to do
left/right arrows	Move cursor
shift+left/right arrows	Select region
ctrl+a	Select all
ctrl+x/c/v	Cut/copy/paste
ctrl+z/y	Undo/redo
ctrl+left/right	Add entry to list or column to matrix
shift+ctrl+left/right	Add copy of current entry/column to to list/matrix
ctrl+up/down	Add row to matrix
shift+ctrl+up/down	Add copy of current row to matrix
ctrl+backspace	Delete current entry in list or column in matrix
ctrl+shift+backspace	Delete current row in matrix

Type...	...to get
norm	$\|\|\blue{[?]}\|\|$
text	$\text{\blue{[?]}}$
sym_name	$\backslash\texttt{\blue{[?]}}$
abs	$\left\|\blue{[?]}\right\|$
sqrt	$\sqrt{\blue{[?]}}$
paren	$\left(\blue{[?]}\right)$
floor	$\lfloor \blue{[?]} \rfloor$
factorial	$\blue{[?]}!$
exp	${\blue{[?]}}^{\blue{[?]}}$
sub	${\blue{[?]}}_{\blue{[?]}}$
frac	$\dfrac{\blue{[?]}}{\blue{[?]}}$
int	$\displaystyle\int{\blue{[?]}}d\blue{[?]}$
defi	$\displaystyle\int_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}d\blue{[?]}$
deriv	$\displaystyle\frac{d}{d\blue{[?]}}\blue{[?]}$
sum	$\displaystyle\sum_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
prod	$\displaystyle\prod_{\blue{[?]}}^{\blue{[?]}}\blue{[?]}$
root	$\sqrt[\blue{[?]}]{\blue{[?]}}$
vec	$\left\langle \blue{[?]} \right\rangle$
mat	$\left(\begin{matrix} \blue{[?]} \end{matrix}\right)$
*	$\cdot$
infinity	$\infty$
arcsin	$\arcsin\left(\blue{[?]}\right)$
arccos	$\arccos\left(\blue{[?]}\right)$
arctan	$\arctan\left(\blue{[?]}\right)$
sin	$\sin\left(\blue{[?]}\right)$
cos	$\cos\left(\blue{[?]}\right)$
tan	$\tan\left(\blue{[?]}\right)$
sec	$\sec\left(\blue{[?]}\right)$
csc	$\csc\left(\blue{[?]}\right)$
cot	$\cot\left(\blue{[?]}\right)$
log	$\log\left(\blue{[?]}\right)$
ln	$\ln\left(\blue{[?]}\right)$
alpha	$\alpha$
beta	$\beta$
gamma	$\gamma$
delta	$\delta$
epsilon	$\epsilon$
zeta	$\zeta$
eta	$\eta$
theta	$\theta$
iota	$\iota$
kappa	$\kappa$
lambda	$\lambda$
mu	$\mu$
nu	$\nu$
xi	$\xi$
omicron	$\omicron$
pi	$\pi$
rho	$\rho$
sigma	$\sigma$
tau	$\tau$
upsilon	$\upsilon$
phi	$\phi$
chi	$\chi$
psi	$\psi$
omega	$\omega$
Gamma	$\Gamma$
Delta	$\Delta$
Theta	$\Theta$
Lambda	$\Lambda$
Xi	$\Xi$
Pi	$\Pi$
Sigma	$\Sigma$
Phi	$\Phi$
Psi	$\Psi$
Omega	$\Omega$

Sample Quiz Questions

Sample Exam Questions

Controls

Symbols

Settings