Exercises: Substitution and Tables

$\newenvironment {prompt}{}{} \newcommand {\ungraded }[0]{} \newcommand {\bigmath }[1]{$\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 substitution and the use of integral tables.

Compute the indefinite integrals below. Since there are many possible answers (which differ by constant values), use the given instructions if needed to choose which possible answer to use.

$\int (12 x+14) \left (3 x^2+7 x-1\right )^5 dx = \answer {\frac {1}{3}(3 x^2+7 x-1)^6}+C$ (Add a constant to your answer if needed so that it equals $1/3$ at $x = 0$ .)

$\int \frac {e^{\sqrt {x}}}{\sqrt {x}} dx = \answer {2 e^{\sqrt {x}}}+C$ (Add a constant to your answer if needed so that it equals $2$ at $x = 0$ .)

$\int \frac {x}{\sqrt {x+3}} dx = \answer {\frac {2}{3} (x-6) \sqrt {x+3}}+C$ (Add a constant to your answer if needed so that it equals $0$ at $x=6$ .)

$\int \frac {\ln |x|}{x} dx = \answer {\frac {1}{2}\ln ^2 |x|}+C$ Remember absolute value in your logarithm. (Add a constant to your answer if needed so that it equals $0$ at $x = 1$ .)

$\int \sin x \sqrt {\cos x} dx = \answer {-\frac {2}{3} \cos ^{\frac {3}{2}}(x)}+C$ (Add a constant to your answer if needed so that it equals $-2/3$ at $x = 0$ .)

$\int \frac {9 (2 x+3)}{3 x^2+9 x+7} dx = \answer {3 \ln \left |3 x^2+9 x+7\right |}+C$ Use absolute values as needed in logarithms. (Add a constant to your answer if needed so that it equals $3 \ln 7$ at $x = 0$ .)

$\int \frac {x}{x^4+81} dx = \answer {\frac {1}{18} \arctan \left (\frac {x^2}{9}\right )}+C$ (Add a constant to your answer if needed so that it equals $0$ at $x = 0$ .)

Make a substitution $x^2 = 9 u$ .

Evaluate the definite integral $\displaystyle \int _{-2}^{-1} (x+1)e^{x^2+2x+1}\ dx.$

Value = $\answer {(1-e)/2}$

Sample Exam Questions

Evaluate the integral $\displaystyle \int _1^3 \left ( x - \sqrt {4 x^2 - 8 x + 13} \right ) dx$ using the fact that $\displaystyle \int _0^4 \sqrt {x^2 + 9} ~ dx = \frac {20 + 9 \ln 3}{2}$ . (Hints will not be displayed until you have chosen a response.)

First complete the square: $4 x^2 - 8 x + 13 = 4 x^2 - 8 x + 4 + (13 - 4) = (2x - 2)^2 + 9.$

Next substitute $u = 2x-2$ (i.e., $x = \frac {u}{2} + 1$ ). Then $du = 2 dx$ and $x=1 \leftrightarrow u = 0$ , $x=3 \leftrightarrow u = 4$ , so $\int _1^3 \left ( x - \sqrt {4x^2 - 8x + 13} \right ) dx = \int _0^4 \left ( \frac {u}{2} + 1 - \sqrt {u^2 + 9} \right ) \frac {du}{2}.$

Now use linearity of the integral to finish: $\int _0^4 \frac {u}{4} ~ du + \int _0^4 \frac {1}{2} ~ du - \int _0^4 \frac {1}{2} \sqrt {u^2+9} ~ du = \left . \frac {u^2}{8} \right |_0^4 + \left . \frac {u}{2} \right |_0^4 - \frac {20 + 9 \ln 3}{4} = 4 - \frac {20+9 \ln 3}{4} = - \frac {4 + 9 \ln 3}{4}.$

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 Exam Questions

Controls

Symbols

Settings