Question 1

Determine whether the improper integral converges or diverges.
-$$\int _ { 1 } ^ { \infty } \frac { 1 } { x ^ { 9 } + 2 } d x$$

Accepted Answer

A)  Diverges 
B)  Converges 
A)  Diverges 
B)  Converges

Question 2

Use a trigonometric substitution to evaluate the integral.
-$$\int \frac { d x } { 2 \sqrt { x } ( 1 + x ) }$$

Accepted Answer

A)  $\frac { 1 } { 2 } \tan ^ { - 1 } \sqrt { x } + C$ 
B)  $\frac { 1 } { 2 } \sin ^ { - 1 } \sqrt { x } + C$ 
C)  $\frac { 1 } { 2 } \ln | x | + C$ 
D)  $\tan ^ { - 1 } \sqrt { x } + C$ 
A)  $\frac { 1 } { 2 } \tan ^ { - 1 } \sqrt { x } + C$ 
B)  $\frac { 1 } { 2 } \sin ^ { - 1 } \sqrt { x } + C$ 
C)  $\frac { 1 } { 2 } \ln | x | + C$ 
D)  $\tan ^ { - 1 } \sqrt { x } + C$

Question 3

Evaluate the integral by first performing long division on the integrand and then writing the proper fraction as a sum of partial fractions.
-$$\int \frac { 3 x ^ { 4 } + 15 x ^ { 2 } + 5 } { x ^ { 3 } + 5 x } d x$$

Accepted Answer

A)  $\frac { 3 } { 2 } x ^ { 2 } - \frac { 1 } { 2 } \ln \left| x ^ { 2 } + 5 \right| + C$ 
B)  $\frac { 3 } { 2 } x ^ { 2 } + \ln | x | - x \ln \left| x ^ { 2 } + 5 \right| + C$ 
C)  $\frac { 3 } { 2 } x ^ { 2 } + \ln | x | + \frac { 1 } { 2 } \ln \left| x ^ { 2 } + 5 \right| + C$ 
D)  $\frac { 3 } { 2 } x ^ { 2 } + \ln | x | - \frac { 1 } { 2 } \ln \left| x ^ { 2 } + 5 \right| + C$ 
A)  $\frac { 3 } { 2 } x ^ { 2 } - \frac { 1 } { 2 } \ln \left| x ^ { 2 } + 5 \right| + C$ 
B)  $\frac { 3 } { 2 } x ^ { 2 } + \ln | x | - x \ln \left| x ^ { 2 } + 5 \right| + C$ 
C)  $\frac { 3 } { 2 } x ^ { 2 } + \ln | x | + \frac { 1 } { 2 } \ln \left| x ^ { 2 } + 5 \right| + C$ 
D)  $\frac { 3 } { 2 } x ^ { 2 } + \ln | x | - \frac { 1 } { 2 } \ln \left| x ^ { 2 } + 5 \right| + C$

Question 4

Evaluate the integral.
-$$\int 20 x \cos \frac { 1 } { 2 } x d x$$

Accepted Answer

A)  $80 \sin \left( \frac { 1 } { 2 } \right) x - 40 x \cos \left( \frac { 1 } { 2 } \right) x + C$ 
B)  $20 x \sin \left( \frac { 1 } { 2 } \right) x - 40 \cos \left( \frac { 1 } { 2 } \right) x + C$ 
C)  $40 x \sin \left( \frac { 1 } { 2 } \right) x + 80 \cos \left( \frac { 1 } { 2 } \right) x + C$ 
D)  $20 \sin \left( \frac { 1 } { 2 } \right) x + 40 x \cos \left( \frac { 1 } { 2 } \right) x + C$ 
A)  $80 \sin \left( \frac { 1 } { 2 } \right) x - 40 x \cos \left( \frac { 1 } { 2 } \right) x + C$ 
B)  $20 x \sin \left( \frac { 1 } { 2 } \right) x - 40 \cos \left( \frac { 1 } { 2 } \right) x + C$ 
C)  $40 x \sin \left( \frac { 1 } { 2 } \right) x + 80 \cos \left( \frac { 1 } { 2 } \right) x + C$ 
D)  $20 \sin \left( \frac { 1 } { 2 } \right) x + 40 x \cos \left( \frac { 1 } { 2 } \right) x + C$

Question 5

Solve the problem.
-Find the length of the curve $y = \ln ( \csc x ) , \pi / 6 \leq x \leq \pi / 2$

Accepted Answer

A)  $\ln ( \sqrt { 3 } + 2 )$ 
B)  $\ln ( \sqrt { 3 } + 1 )$ 
C)  $\ln ( \sqrt { 3 } )$ 
D)  $1 - \ln ( \sqrt { 3 } + 2 )$ 
A)  $\ln ( \sqrt { 3 } + 2 )$ 
B)  $\ln ( \sqrt { 3 } + 1 )$ 
C)  $\ln ( \sqrt { 3 } )$ 
D)  $1 - \ln ( \sqrt { 3 } + 2 )$

Question 6

Use Simpson's Rule with n = 4 steps to estimate the integral.
-$$\int _ { - \pi } ^ { 0 } \sin x d x$$

Accepted Answer

A)  $- \frac { 1 + 2 \sqrt { 2 } } { 6 } \pi$ 
B)  $- ( 1 + 2 \sqrt { 2 } ) \pi$ 
C)  $- \frac { 1 + \sqrt { 2 } } { 4 } \pi$ 
D)  $- \frac { 2 + \sqrt { 2 } } { 6 } \pi$ 
A)  $- \frac { 1 + 2 \sqrt { 2 } } { 6 } \pi$ 
B)  $- ( 1 + 2 \sqrt { 2 } ) \pi$ 
C)  $- \frac { 1 + \sqrt { 2 } } { 4 } \pi$ 
D)  $- \frac { 2 + \sqrt { 2 } } { 6 } \pi$

Question 7

Evaluate the integral.
-$$\int _ { - \pi } ^ { \pi } \sqrt { \frac { 1 + \cos x } { 2 } } d x$$

Accepted Answer

A)  4 
B)  1 
C)  0 
D)  2 
A)  4 
B)  1 
C)  0 
D)  2

Question 8

Express the integrand as a sum of partial fractions and evaluate the integral.
-$$\int _ { 5 } ^ { 9 } \frac { 6 } { x ^ { 2 } - 9 } d x$$

Accepted Answer

A)  -0.693 
B)  2.773 
C)  3.693 
D)  0.693 
A)  -0.693 
B)  2.773 
C)  3.693 
D)  0.693

Question 9

Evaluate the integral.
-$$\int \cos 6 x \cos 4 x d x$$

Accepted Answer

A)  $\frac { 1 } { 4 } \sin 2 x + \frac { 1 } { 20 } \sin 10 x + C$ 
B)  $\frac { 1 } { 4 } \sin 2 x - \frac { 1 } { 20 } \sin 10 x + C$ 
C)  $\frac { 1 } { 4 } \sin 6 x + \frac { 1 } { 20 } \sin 4 x + C$ 
D)  $\frac { 1 } { 4 } \cos 2 x + \frac { 1 } { 20 } \cos 10 x + C$ 
A)  $\frac { 1 } { 4 } \sin 2 x + \frac { 1 } { 20 } \sin 10 x + C$ 
B)  $\frac { 1 } { 4 } \sin 2 x - \frac { 1 } { 20 } \sin 10 x + C$ 
C)  $\frac { 1 } { 4 } \sin 6 x + \frac { 1 } { 20 } \sin 4 x + C$ 
D)  $\frac { 1 } { 4 } \cos 2 x + \frac { 1 } { 20 } \cos 10 x + C$

Question 10

Use the Trapezoidal Rule with n = 4 steps to estimate the integral.
-$$\int _ { - 1 } ^ { 0 } \sin \pi t d t$$

Accepted Answer

A)  $- \frac { 1 + \sqrt { 2 } } { 8 }$ 
B)  $- \frac { 1 + \sqrt { 2 } } { 2 }$ 
C)  $- \frac { 1 + \sqrt { 2 } } { 4 }$ 
D)  $- 1 - \sqrt { 2 }$ 
A)  $- \frac { 1 + \sqrt { 2 } } { 8 }$ 
B)  $- \frac { 1 + \sqrt { 2 } } { 2 }$ 
C)  $- \frac { 1 + \sqrt { 2 } } { 4 }$ 
D)  $- 1 - \sqrt { 2 }$

Question 11

Provide an appropriate response.
-$\text { A student needs } \int _ { - \infty } ^ { + \infty } \mathrm { e } ^ { - | \mathrm { x } | } \mathrm { d } x \text {. Is this integral the same as } 2 \int _ { 0 } ^ { + \infty } \mathrm { e } ^ { - | \mathrm { x } | } \mathrm { dx } \text {, and if so, why? }$

Accepted Answer

Yes
the fu

Question 12

Evaluate the improper integral.
-$$\int _ { - 8 } ^ { 1 } \frac { d x } { x ^ { 2 / 3 } }$$

Accepted Answer

A)  -3 
B)  3 
C)  0 
D)  9 
A)  -3 
B)  3 
C)  0 
D)  9

Question 13

Solve the problem.
-The height of a vase is 5 inches. The table shows the circumference of the vase (in inches) at half-inch intervals starting from the top down. Estimate the volume of the vase by using the Trapezoidal rule with $n = 10$. Round your answer to the nearest thousandth. [Hint: you will first need to find the areas of the cross-sections that correspond to the given circumferences.]
$\begin{array}{lr}
{\text { Circumferences }} \
\hline 4.7 & 8.1 \
4.2 & 9.4 \
4.1 & 10.1 \
4.8 & 8.5 \
5.6 & 6.4 \
6.8 &
\end{array}$

Accepted Answer

A)  $61.856 \mathrm { in } ^ { 3 }$ 
B)  $62.876 \mathrm { in } ^ { 3 }$ 
C)  $101.552$ in. 3 
D)  $105.479$ in. $^ { 3 }$ 
A)  $61.856 \mathrm { in } ^ { 3 }$ 
B)  $62.876 \mathrm { in } ^ { 3 }$ 
C)  $101.552$ in. 3 
D)  $105.479$ in. $^ { 3 }$

Question 14

Provide an appropriate response.
-$\text { A student knows that } \int _ { a } ^ { + \infty } f ( x ) d x \text { converges. Does } \int _ { - \infty } ^ { a } \mathrm { f } ( \mathrm { x } ) \mathrm { dx } \text { also necessarily converge? }$

Accepted Answer

A) True 
 B)False

Question 15

Expand the quotient by partial fractions.
-$$\frac { 2 x + 32 } { ( x + 1 ) ( x + 7 ) }$$

Accepted Answer

A)  $\frac { 5 } { x + 1 } + \frac { - 3 } { x + 7 }$ 
B)  $\frac { 5 } { x - 1 } + \frac { - 3 } { x - 7 }$ 
C)  $\frac { 5 } { x + 1 } + \frac { 3 } { x + 7 }$ 
D)  $\frac { 3 } { x + 1 } + \frac { - 5 } { x + 7 }$ 
A)  $\frac { 5 } { x + 1 } + \frac { - 3 } { x + 7 }$ 
B)  $\frac { 5 } { x - 1 } + \frac { - 3 } { x - 7 }$ 
C)  $\frac { 5 } { x + 1 } + \frac { 3 } { x + 7 }$ 
D)  $\frac { 3 } { x + 1 } + \frac { - 5 } { x + 7 }$

Question 16

Use reduction formulas to evaluate the integral.
-$$\int \sec ^ { 3 } 4 x d x$$

Accepted Answer

A)  $\frac { 1 } { 8 } \sec ^ { 2 } 4 x \tan 4 x + \frac { 1 } { 8 } \ln | \sec 4 x + \tan 4 x | + C$ 
B)  $\frac { 1 } { 8 } \sec 4 x \tan 4 x + \frac { 1 } { 8 } \ln | \sec 4 x + \tan 4 x | + C$ 
C)  $\frac { 1 } { 8 } \sec 4 x \tan 4 x + \frac { 1 } { 8 } \ln | \sec 4 x + \cot 4 x | + C$ 
D)  $\frac { 1 } { 8 } \sec 4 x \tan 4 x - \frac { 1 } { 4 } \ln | \sec 4 x + \tan 4 x | + C$ 
A)  $\frac { 1 } { 8 } \sec ^ { 2 } 4 x \tan 4 x + \frac { 1 } { 8 } \ln | \sec 4 x + \tan 4 x | + C$ 
B)  $\frac { 1 } { 8 } \sec 4 x \tan 4 x + \frac { 1 } { 8 } \ln | \sec 4 x + \tan 4 x | + C$ 
C)  $\frac { 1 } { 8 } \sec 4 x \tan 4 x + \frac { 1 } { 8 } \ln | \sec 4 x + \cot 4 x | + C$ 
D)  $\frac { 1 } { 8 } \sec 4 x \tan 4 x - \frac { 1 } { 4 } \ln | \sec 4 x + \tan 4 x | + C$

Question 17

Solve the initial value problem for x as a function of t.
-$$\left( 2 t ^ { 3 } - 2 t ^ { 2 } + t - 1 \right) \frac { d x } { d t } = 3 , x ( 2 ) = 0$$

Accepted Answer

A)  $x = \ln | t - 1 | - \frac { 1 } { 2 } \ln \left| t ^ { 2 } + \frac { 1 } { 2 } \right| + \frac { 1 } { 2 } \ln 4.5$ 
B)  $x = \ln | t - 1 | - \sqrt { 2 } \tan ^ { - 1 } \sqrt { 2 } t + \sqrt { 2 } \tan ^ { - 1 } 2 \sqrt { 2 }$ 
C)  $x = \ln | t - 1 | - \tan ^ { - 1 } \sqrt { 2 } t - \ln \left| t ^ { 2 } + \frac { 1 } { 2 } \right| + \tan ^ { - 1 } 2 \sqrt { 2 } + \ln 4.5$ 
D)  $x = \ln | t - 1 | - \sqrt { 2 } \tan ^ { - 1 } \sqrt { 2 } t - \frac { 1 } { 2 } \ln \left| t ^ { 2 } + \frac { 1 } { 2 } \right| + \sqrt { 2 } \tan ^ { - 1 } 2 \sqrt { 2 } + \frac { 1 } { 2 } \ln 4.5$ 
A)  $x = \ln | t - 1 | - \frac { 1 } { 2 } \ln \left| t ^ { 2 } + \frac { 1 } { 2 } \right| + \frac { 1 } { 2 } \ln 4.5$ 
B)  $x = \ln | t - 1 | - \sqrt { 2 } \tan ^ { - 1 } \sqrt { 2 } t + \sqrt { 2 } \tan ^ { - 1 } 2 \sqrt { 2 }$ 
C)  $x = \ln | t - 1 | - \tan ^ { - 1 } \sqrt { 2 } t - \ln \left| t ^ { 2 } + \frac { 1 } { 2 } \right| + \tan ^ { - 1 } 2 \sqrt { 2 } + \ln 4.5$ 
D)  $x = \ln | t - 1 | - \sqrt { 2 } \tan ^ { - 1 } \sqrt { 2 } t - \frac { 1 } { 2 } \ln \left| t ^ { 2 } + \frac { 1 } { 2 } \right| + \sqrt { 2 } \tan ^ { - 1 } 2 \sqrt { 2 } + \frac { 1 } { 2 } \ln 4.5$

Question 18

Use integration by parts to establish a reduction formula for the integral.
-$$\int \mathrm { x } ^ { \mathrm { n } } \mathrm { e } ^ { - a x } \mathrm { dx }$$

Accepted Answer

A)  $\int x ^ { n } e ^ { - a x } d x = \frac { x ^ { n } e ^ { - a x } } { a } - \frac { n } { a } \int x ^ { n - 1 } e ^ { - a x } d x$ 
B)  $\int x ^ { n } e ^ { - a x } d x = - a x ^ { n } e ^ { - a x } + n a \int x ^ { n - 1 } e ^ { - a x } d x$ 
C)  $\int x ^ { n } e ^ { - a x } d x = - \frac { x ^ { n } e ^ { - a x } } { a } + \frac { n } { a } \int x ^ { n - 2 } e ^ { - a x } d x$ 
D)  $\int x ^ { n } e ^ { - a x } d x = - \frac { x ^ { n } e ^ { - a x } } { a } + \frac { n } { a } \int x ^ { n - 1 } e ^ { - a x } d x$ 
A)  $\int x ^ { n } e ^ { - a x } d x = \frac { x ^ { n } e ^ { - a x } } { a } - \frac { n } { a } \int x ^ { n - 1 } e ^ { - a x } d x$ 
B)  $\int x ^ { n } e ^ { - a x } d x = - a x ^ { n } e ^ { - a x } + n a \int x ^ { n - 1 } e ^ { - a x } d x$ 
C)  $\int x ^ { n } e ^ { - a x } d x = - \frac { x ^ { n } e ^ { - a x } } { a } + \frac { n } { a } \int x ^ { n - 2 } e ^ { - a x } d x$ 
D)  $\int x ^ { n } e ^ { - a x } d x = - \frac { x ^ { n } e ^ { - a x } } { a } + \frac { n } { a } \int x ^ { n - 1 } e ^ { - a x } d x$

Question 19

Evaluate the integral.
-$$\int \frac { \cot ^ { 3 } x } { 7 } d x$$

Accepted Answer

A)  $\frac { 1 } { 14 } \cot ^ { 2 } x + \ln | \sin x | + C$ 
B)  $- \frac { 1 } { 14 } \cot ^ { 2 } x - \frac { 1 } { 7 } \ln | \sin x | + C$ 
C)  $\frac { 1 } { 28 } \cot ^ { 4 } x + C$ 
D)  $\frac { 1 } { 28 } \cot ^ { 4 } x \sec x + C$ 
A)  $\frac { 1 } { 14 } \cot ^ { 2 } x + \ln | \sin x | + C$ 
B)  $- \frac { 1 } { 14 } \cot ^ { 2 } x - \frac { 1 } { 7 } \ln | \sin x | + C$ 
C)  $\frac { 1 } { 28 } \cot ^ { 4 } x + C$ 
D)  $\frac { 1 } { 28 } \cot ^ { 4 } x \sec x + C$

Question 20

Find the value of the constant k so that the given function in a probability density function for a random variable over the
specified interval.
-$f ( x ) = k ( 19 - x )$ over $[ 0,19 ]$

Accepted Answer

A)  $\frac { 1 } { 361 }$ 
B)  361 
C)  $\frac { 2 } { 361 }$ 
D)  19 
A)  $\frac { 1 } { 361 }$ 
B)  361 
C)  $\frac { 2 } { 361 }$ 
D)  19

Determine whether the improper integral converges or diverges. - $\int _ { 1 } ^ { \infty } \frac { 1 } { x ^ { 9 } + 2 } d x$

Use a trigonometric substitution to evaluate the integral. - $\int \frac { d x } { 2 \sqrt { x } ( 1 + x ) }$

Evaluate the integral by first performing long division on the integrand and then writing the proper fraction as a sum of partial fractions. - $\int \frac { 3 x ^ { 4 } + 15 x ^ { 2 } + 5 } { x ^ { 3 } + 5 x } d x$

Evaluate the integral. - $\int 20 x \cos \frac { 1 } { 2 } x d x$

Solve the problem. -Find the length of the curve $y = \ln ( \csc x ) , \pi / 6 \leq x \leq \pi / 2$

Use Simpson's Rule with n = 4 steps to estimate the integral. - $\int _ { - \pi } ^ { 0 } \sin x d x$

Evaluate the integral. - $\int _ { - \pi } ^ { \pi } \sqrt { \frac { 1 + \cos x } { 2 } } d x$

Express the integrand as a sum of partial fractions and evaluate the integral. - $\int _ { 5 } ^ { 9 } \frac { 6 } { x ^ { 2 } - 9 } d x$

Evaluate the integral. - $\int \cos 6 x \cos 4 x d x$

Use the Trapezoidal Rule with n = 4 steps to estimate the integral. - $\int _ { - 1 } ^ { 0 } \sin \pi t d t$

Evaluate the improper integral. - $\int _ { - 8 } ^ { 1 } \frac { d x } { x ^ { 2 / 3 } }$

Provide an appropriate response. - $\text { A student knows that } \int _ { a } ^ { + \infty } f ( x ) d x \text { converges. Does } \int _ { - \infty } ^ { a } \mathrm { f } ( \mathrm { x } ) \mathrm { dx } \text { also necessarily converge? }$

Expand the quotient by partial fractions. - $\frac { 2 x + 32 } { ( x + 1 ) ( x + 7 ) }$

Use reduction formulas to evaluate the integral. - $\int \sec ^ { 3 } 4 x d x$

Solve the initial value problem for x as a function of t. - $\left( 2 t ^ { 3 } - 2 t ^ { 2 } + t - 1 \right) \frac { d x } { d t } = 3 , x ( 2 ) = 0$

Use integration by parts to establish a reduction formula for the integral. - $\int \mathrm { x } ^ { \mathrm { n } } \mathrm { e } ^ { - a x } \mathrm { dx }$

Evaluate the integral. - $\int \frac { \cot ^ { 3 } x } { 7 } d x$

Find the value of the constant k so that the given function in a probability density function for a random variable over the specified interval. - $f ( x ) = k ( 19 - x )$ over $[ 0,19 ]$

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Exam 9: Techniques of Integration

Determine whether the improper integral converges or diverges. - ∫1∞1x9+2dx\int _ { 1 } ^ { \infty } \frac { 1 } { x ^ { 9 } + 2 } d x∫1∞​x9+21​dx

Use a trigonometric substitution to evaluate the integral. - ∫dx2x(1+x)\int \frac { d x } { 2 \sqrt { x } ( 1 + x ) }∫2x​(1+x)dx​

Evaluate the integral by first performing long division on the integrand and then writing the proper fraction as a sum of partial fractions. - ∫3x4+15x2+5x3+5xdx\int \frac { 3 x ^ { 4 } + 15 x ^ { 2 } + 5 } { x ^ { 3 } + 5 x } d x∫x3+5x3x4+15x2+5​dx

Evaluate the integral. - ∫20xcos⁡12xdx\int 20 x \cos \frac { 1 } { 2 } x d x∫20xcos21​xdx

Solve the problem. -Find the length of the curve y=ln⁡(csc⁡x),π/6≤x≤π/2y = \ln ( \csc x ) , \pi / 6 \leq x \leq \pi / 2y=ln(cscx),π/6≤x≤π/2

Use Simpson's Rule with n = 4 steps to estimate the integral. - ∫−π0sin⁡xdx\int _ { - \pi } ^ { 0 } \sin x d x∫−π0​sinxdx

Evaluate the integral. - ∫−ππ1+cos⁡x2dx\int _ { - \pi } ^ { \pi } \sqrt { \frac { 1 + \cos x } { 2 } } d x∫−ππ​21+cosx​​dx

Express the integrand as a sum of partial fractions and evaluate the integral. - ∫596x2−9dx\int _ { 5 } ^ { 9 } \frac { 6 } { x ^ { 2 } - 9 } d x∫59​x2−96​dx

Evaluate the integral. - ∫cos⁡6xcos⁡4xdx\int \cos 6 x \cos 4 x d x∫cos6xcos4xdx

Use the Trapezoidal Rule with n = 4 steps to estimate the integral. - ∫−10sin⁡πtdt\int _ { - 1 } ^ { 0 } \sin \pi t d t∫−10​sinπtdt

Evaluate the improper integral. - ∫−81dxx2/3\int _ { - 8 } ^ { 1 } \frac { d x } { x ^ { 2 / 3 } }∫−81​x2/3dx​

Expand the quotient by partial fractions. - 2x+32(x+1)(x+7)\frac { 2 x + 32 } { ( x + 1 ) ( x + 7 ) }(x+1)(x+7)2x+32​

Use reduction formulas to evaluate the integral. - ∫sec⁡34xdx\int \sec ^ { 3 } 4 x d x∫sec34xdx

Solve the initial value problem for x as a function of t. - (2t3−2t2+t−1)dxdt=3,x(2)=0\left( 2 t ^ { 3 } - 2 t ^ { 2 } + t - 1 \right) \frac { d x } { d t } = 3 , x ( 2 ) = 0(2t3−2t2+t−1)dtdx​=3,x(2)=0

Use integration by parts to establish a reduction formula for the integral. - ∫xne−axdx\int \mathrm { x } ^ { \mathrm { n } } \mathrm { e } ^ { - a x } \mathrm { dx }∫xne−axdx

Evaluate the integral. - ∫cot⁡3x7dx\int \frac { \cot ^ { 3 } x } { 7 } d x∫7cot3x​dx

Find the value of the constant k so that the given function in a probability density function for a random variable over the specified interval. - f(x)=k(19−x)f ( x ) = k ( 19 - x )f(x)=k(19−x) over [0,19][ 0,19 ][0,19]