Question 1

Use the Trapezoidal Rule with n = 4 steps to estimate the integral.
-$$\int _ { 0 } ^ { 2 } \left( x ^ { 4 } + 8 \right) d x$$

Accepted Answer

A)  $\frac { 497 } { 16 }$ 
B)  $\frac { 269 } { 12 }$ 
C)  $\frac { 369 } { 16 }$ 
D)  $\frac { 369 } { 8 }$ 
A)  $\frac { 497 } { 16 }$ 
B)  $\frac { 269 } { 12 }$ 
C)  $\frac { 369 } { 16 }$ 
D)  $\frac { 369 } { 8 }$

Question 2

Express the integrand as a sum of partial fractions and evaluate the integral.
-$$\int \frac { 2 x ^ { 2 } + x + 1 } { \left( x ^ { 2 } + 3 \right) ( x - 5 ) } d x$$

Accepted Answer

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

Question 3

Evaluate the improper integral or state that it is divergent.
-$$\int _ { 5 } ^ { \infty } \frac { d t } { t ^ { 2 } - 4 t }$$

Accepted Answer

A)  $- \frac { 1 } { 4 } \ln 5$ 
B)  $\frac { 1 } { 5 } \ln 4$ 
C)  $\frac { 1 } { 4 } \ln 5$ 
D)  $4 \ln 5$ 
A)  $- \frac { 1 } { 4 } \ln 5$ 
B)  $\frac { 1 } { 5 } \ln 4$ 
C)  $\frac { 1 } { 4 } \ln 5$ 
D)  $4 \ln 5$

Question 4

Solve the problem.
-Suppose that the accompanying table shows the velocity of a car every second for 8 seconds. Use the Trapezoidal Rule to approximate the distance traveled by the car in the 8 seconds. Round your answer to the nearest tenth if necessary.
$\begin{array}{l|l}
\text { Time (sec) } & \text { Velocity (ft/sec) } \
\hline 0 & 17 \
1 & 18 \
2 & 19 \
3 & 21 \
4 & 20 \
5 & 22 \
6 & 19 \
7 & 17 \
8 & 18
\end{array}$

Accepted Answer

A)  153.5 ft 
B)  171 ft 
C)  233.5 ft 
D)  307 ft 
A)  153.5 ft 
B)  171 ft 
C)  233.5 ft 
D)  307 ft

Question 5

Evaluate the improper integral.
-$$\int _ { 0 } ^ { 49 } \frac { d x } { \sqrt { 49 - x } }$$

Accepted Answer

A)  7 
B)  0 
C)  14 
D)  -14 
A)  7 
B)  0 
C)  14 
D)  -14

Question 6

Evaluate the integral.
-$$\int _ { \pi / 12 } ^ { \pi / 6 } \cot ^ { 4 } 3 t d t$$

Accepted Answer

A)  $\frac { \pi } { 12 } - \frac { 2 } { 9 }$ 
B)  $\frac { \pi } { 12 } - \frac { 2 } { 3 }$ 
C)  $\frac { \pi } { 6 } - \frac { 2 } { 9 }$ 
D)  $- \frac { 2 } { 9 }$ 
A)  $\frac { \pi } { 12 } - \frac { 2 } { 9 }$ 
B)  $\frac { \pi } { 12 } - \frac { 2 } { 3 }$ 
C)  $\frac { \pi } { 6 } - \frac { 2 } { 9 }$ 
D)  $- \frac { 2 } { 9 }$

Question 7

Provide an appropriate response.
-The standard normal probability density function is defined by $f ( x ) = \frac { 1 } { \sqrt { 2 \pi } } e ^ { - x ^ { 2 } / 2 }$.
(a) Use the fact that $\int _ { - \infty } ^ { \infty } \frac { 1 } { \sqrt { 2 \pi } } e ^ { - } x ^ { 2 } / 2 d x = 1$ to show that $\int _ { 0 } ^ { \infty } \frac { 1 } { \sqrt { 2 \pi } } x ^ { 2 } e ^ { - x ^ { 2 } / 2 } \mathrm { dx } = \frac { 1 } { 2 }$.
(b) Use the result in part (a) to show that the standard normal probability density function has variance $1 .$

Accepted Answer

(a) Since @#LAT-DLM& and @#LAT-DLM& is a

Question 8

Integrate the function.
-$$\int \frac { d x } { x \sqrt { 16 x ^ { 2 } - 36 } }$$

Accepted Answer

A)  $\frac { 2 } { 3 } \sec ^ { - 1 } \frac { 2 } { 3 } x + C$ 
B)  $\frac { 2 } { 3 } \sin ^ { - 1 } \frac { 2 } { 3 } x + C$ 
C)  $\frac { 1 } { 6 } \sec ^ { - 1 } \frac { 2 } { 3 } x + C$ 
D)  $\frac { 1 } { 4 } \sin ^ { - 1 } \frac { 2 } { 3 } x + C$ 
A)  $\frac { 2 } { 3 } \sec ^ { - 1 } \frac { 2 } { 3 } x + C$ 
B)  $\frac { 2 } { 3 } \sin ^ { - 1 } \frac { 2 } { 3 } x + C$ 
C)  $\frac { 1 } { 6 } \sec ^ { - 1 } \frac { 2 } { 3 } x + C$ 
D)  $\frac { 1 } { 4 } \sin ^ { - 1 } \frac { 2 } { 3 } x + C$

Question 9

Solve the problem.
-Find the volume of the solid generated by revolving the region in the first quadrant bounded by $y = e ^ { x }$ and the $x$-axis, from $x = 0$ to $x = \ln 3$, about the $y$-axis.

Accepted Answer

A)  $6 \pi \ln 3$ 
B)  $2 \pi ( 3 \ln 3 - 2 )$ 
C)  $2 \pi ( 3 \ln 3 - 3 )$ 
D)  $3 \pi \ln 3$ 
A)  $6 \pi \ln 3$ 
B)  $2 \pi ( 3 \ln 3 - 2 )$ 
C)  $2 \pi ( 3 \ln 3 - 3 )$ 
D)  $3 \pi \ln 3$

Question 10

Evaluate the integral.
-$\int 8 x e ^ { x } d x$

Accepted Answer

A)  $8 x e ^ { x } - 8 e ^ { x } + C$ 
B)  $x e ^ { x } - 8 e ^ { x } + C$ 
C)  $8 e ^ { x } - e ^ { x} + C$ 
D)  $8 \mathrm { e } ^ { \mathrm { x } } - 8 x \mathrm { e } ^ { \mathrm { x } } + \mathrm { C }$ 
A)  $8 x e ^ { x } - 8 e ^ { x } + C$ 
B)  $x e ^ { x } - 8 e ^ { x } + C$ 
C)  $8 e ^ { x } - e ^ { x} + C$ 
D)  $8 \mathrm { e } ^ { \mathrm { x } } - 8 x \mathrm { e } ^ { \mathrm { x } } + \mathrm { C }$

Question 11

Provide an appropriate response.
-Let $f ( x ) = \frac { 1 } { x ^ { 2 } } , x \geq 1$
(a) Find the area of the region between the graph of $y = f ( x )$ and the $x$-axis.
(b) If the region described in part (a) is revolved about the x-axis, what is the volume of the solid that is generated?
(c) A surface is generated by revolving the graph of $y = f ( x )$ about the $x$-axis. Write an integral expression for the surface area and show that the integral converges.
(d) Use numerical techniques to estimate the area of the region in part (c), to an accuracy of at least two decimal places.

Accepted Answer

(a) 1
(b) @#LAT-DLM&
(c) @#LAT-DLM& conv

Question 12

Evaluate the improper integral or state that it is divergent.
-$$\int _ { - \infty } ^ { 0 } \frac { d x } { ( 25 + x ) \sqrt { x } }$$

Accepted Answer

A)  $\frac { \pi } { 5 }$ 
B)  0 
C)  $- \frac { \pi } { 5 }$ 
D)  $- 5 \pi$ 
A)  $\frac { \pi } { 5 }$ 
B)  0 
C)  $- \frac { \pi } { 5 }$ 
D)  $- 5 \pi$

Question 13

Find the surface area or volume.
-Use an integral table and a calculator to find to two decimal places the area of the surface generated by revolving the curve $y = x ^ { 2 } , - 1 \leq x \leq 3$, about the $x$-axis.

Accepted Answer

A)  189.04 
B)  265.13 
C)  324.07 
D)  378.08 
A)  189.04 
B)  265.13 
C)  324.07 
D)  378.08

Question 14

Use Simpson's Rule with n = 4 steps to estimate the integral.
-$$\int _ { 0 } ^ { 2 } \left( x ^ { 4 } + 4 \right) d x$$

Accepted Answer

A)  $\frac { 173 } { 24 }$ 
B)  $\frac { 281 } { 24 }$ 
C)  $\frac { 241 } { 16 }$ 
D)  $\frac { 173 } { 12 }$ 
A)  $\frac { 173 } { 24 }$ 
B)  $\frac { 281 } { 24 }$ 
C)  $\frac { 241 } { 16 }$ 
D)  $\frac { 173 } { 12 }$

Question 15

Evaluate the improper integral.
-$$\int _ { 9 } ^ { 18 / \sqrt { 3 } } \frac { d t } { t \sqrt { t ^ { 2 } - 81 } }$$

Accepted Answer

A)  $\frac { \pi } { 18 }$ 
B)  $\frac { 1 } { 9 }$ 
C)  $\frac { \pi } { 54 }$ 
D)  $\frac { \pi } { 3 }$ 
A)  $\frac { \pi } { 18 }$ 
B)  $\frac { 1 } { 9 }$ 
C)  $\frac { \pi } { 54 }$ 
D)  $\frac { \pi } { 3 }$

Question 16

Find the surface area or volume.
-The region between the curve $y = \sin x , 0 \leq x \leq 1.9$, and the $x$-axis is revolved about the $x$-axis to generate a solid. Use a table of integrals to find, to two decimal places, the volume of the solid generated.

Accepted Answer

A)  3.66 
B)  3.47 
C)  3.53 
D)  3.31 
A)  3.66 
B)  3.47 
C)  3.53 
D)  3.31

Question 17

Evaluate the integral.
-$$\int _ { 0 } ^ { \pi } \sqrt { 1 - \cos ^ { 2 } x } d x$$

Accepted Answer

A)  $\frac { \sqrt { 2 } } { 2 }$ 
B)  0 
C)  2 
D)  $\sqrt { 2 }$ 
A)  $\frac { \sqrt { 2 } } { 2 }$ 
B)  0 
C)  2 
D)  $\sqrt { 2 }$

Question 18

Evaluate the integral.
-$$\int x \sin ^ { - 1 } x d x$$

Accepted Answer

A)  $\frac { x ^ { 2 } } { 2 } \sin ^ { - 1 } x - \frac { 1 } { 4 } x \sin ^ { - 1 } x - \frac { 1 } { 4 } \sqrt { 1 - x ^ { 2 } } + C$ 
B)  $\frac { x ^ { 2 } } { 2 } \sin ^ { - 1 } x + \frac { 1 } { 2 } \sin ^ { - 1 } x - \frac { 1 } { 2 } x \sqrt { 1 - x ^ { 2 } } + C$ 
C)  $\frac { x ^ { 2 } } { 2 } \sin ^ { - 1 } x - \frac { 1 } { 4 } \sin ^ { - 1 } x + \frac { 1 } { 4 } x \sqrt { 1 - x ^ { 2 } } + C$ 
D)  $\frac { x ^ { 2 } } { 2 } \sin ^ { - 1 } x - \frac { 1 } { 4 } \cos ^ { - 1 } x + \frac { 1 } { 4 } \sqrt { 1 - x ^ { 2 } } + C$ 
A)  $\frac { x ^ { 2 } } { 2 } \sin ^ { - 1 } x - \frac { 1 } { 4 } x \sin ^ { - 1 } x - \frac { 1 } { 4 } \sqrt { 1 - x ^ { 2 } } + C$ 
B)  $\frac { x ^ { 2 } } { 2 } \sin ^ { - 1 } x + \frac { 1 } { 2 } \sin ^ { - 1 } x - \frac { 1 } { 2 } x \sqrt { 1 - x ^ { 2 } } + C$ 
C)  $\frac { x ^ { 2 } } { 2 } \sin ^ { - 1 } x - \frac { 1 } { 4 } \sin ^ { - 1 } x + \frac { 1 } { 4 } x \sqrt { 1 - x ^ { 2 } } + C$ 
D)  $\frac { x ^ { 2 } } { 2 } \sin ^ { - 1 } x - \frac { 1 } { 4 } \cos ^ { - 1 } x + \frac { 1 } { 4 } \sqrt { 1 - x ^ { 2 } } + C$

Question 19

Evaluate the improper integral or state that it is divergent.
-$$\int _ { 0 } ^ { \infty } \frac { 2 d x } { 64 + x ^ { 2 } }$$

Accepted Answer

A)  $\pi + 8$ 
B)  $\frac { \pi } { 64 }$ 
C)  0 
D)  $\frac { \pi } { 8 }$ 
A)  $\pi + 8$ 
B)  $\frac { \pi } { 64 }$ 
C)  0 
D)  $\frac { \pi } { 8 }$

Question 20

Use reduction formulas to evaluate the integral.
-$\int 3 \cos ^ { 3 } x \sin ^ { 6 } x d x$

Accepted Answer

A)  $\frac { 3 } { 7 } \left( \sin ^ { 7 } x - \sin ^ { 9 } x \right) + C$ 
B)  $\frac { 3 } { 5 } \cos ^ { 5 } x - \frac { 3 } { 7 } \cos ^ { 7 } x + C$ 
C)  $\frac { 3 } { 7 } \sin ^ { 7 } x - \frac { 1 } { 3 } \sin ^ { 9 } x + C$ 
D)  $\frac { 3 } { 5 } \sin ^ { 5 } x - \frac { 1 } { 3 } \cos ^ { 9 } x + C$ 
A)  $\frac { 3 } { 7 } \left( \sin ^ { 7 } x - \sin ^ { 9 } x \right) + C$ 
B)  $\frac { 3 } { 5 } \cos ^ { 5 } x - \frac { 3 } { 7 } \cos ^ { 7 } x + C$ 
C)  $\frac { 3 } { 7 } \sin ^ { 7 } x - \frac { 1 } { 3 } \sin ^ { 9 } x + C$ 
D)  $\frac { 3 } { 5 } \sin ^ { 5 } x - \frac { 1 } { 3 } \cos ^ { 9 } x + C$

Use the Trapezoidal Rule with n = 4 steps to estimate the integral. - $\int _ { 0 } ^ { 2 } \left( x ^ { 4 } + 8 \right) d x$

Express the integrand as a sum of partial fractions and evaluate the integral. - $\int \frac { 2 x ^ { 2 } + x + 1 } { \left( x ^ { 2 } + 3 \right) ( x - 5 ) } d x$

Evaluate the improper integral or state that it is divergent. - $\int _ { 5 } ^ { \infty } \frac { d t } { t ^ { 2 } - 4 t }$

Evaluate the improper integral. - $\int _ { 0 } ^ { 49 } \frac { d x } { \sqrt { 49 - x } }$

Evaluate the integral. - $\int _ { \pi / 12 } ^ { \pi / 6 } \cot ^ { 4 } 3 t d t$

Integrate the function. - $\int \frac { d x } { x \sqrt { 16 x ^ { 2 } - 36 } }$

Solve the problem. -Find the volume of the solid generated by revolving the region in the first quadrant bounded by $y = e ^ { x }$ and the $x$ -axis, from $x = 0$ to $x = \ln 3$ , about the $y$ -axis.

Evaluate the integral. - $\int 8 x e ^ { x } d x$

Evaluate the improper integral or state that it is divergent. - $\int _ { - \infty } ^ { 0 } \frac { d x } { ( 25 + x ) \sqrt { x } }$

Find the surface area or volume. -Use an integral table and a calculator to find to two decimal places the area of the surface generated by revolving the curve $y = x ^ { 2 } , - 1 \leq x \leq 3$ , about the $x$ -axis.

Use Simpson's Rule with n = 4 steps to estimate the integral. - $\int _ { 0 } ^ { 2 } \left( x ^ { 4 } + 4 \right) d x$

Evaluate the improper integral. - $\int _ { 9 } ^ { 18 / \sqrt { 3 } } \frac { d t } { t \sqrt { t ^ { 2 } - 81 } }$

Find the surface area or volume. -The region between the curve $y = \sin x , 0 \leq x \leq 1.9$ , and the $x$ -axis is revolved about the $x$ -axis to generate a solid. Use a table of integrals to find, to two decimal places, the volume of the solid generated.

Evaluate the integral. - $\int _ { 0 } ^ { \pi } \sqrt { 1 - \cos ^ { 2 } x } d x$

Evaluate the integral. - $\int x \sin ^ { - 1 } x d x$

Evaluate the improper integral or state that it is divergent. - $\int _ { 0 } ^ { \infty } \frac { 2 d x } { 64 + x ^ { 2 } }$

Use reduction formulas to evaluate the integral. - $\int 3 \cos ^ { 3 } x \sin ^ { 6 } x d x$

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Exam 9: Techniques of Integration

Use the Trapezoidal Rule with n = 4 steps to estimate the integral. - ∫02(x4+8)dx\int _ { 0 } ^ { 2 } \left( x ^ { 4 } + 8 \right) d x∫02​(x4+8)dx

Express the integrand as a sum of partial fractions and evaluate the integral. - ∫2x2+x+1(x2+3)(x−5)dx\int \frac { 2 x ^ { 2 } + x + 1 } { \left( x ^ { 2 } + 3 \right) ( x - 5 ) } d x∫(x2+3)(x−5)2x2+x+1​dx

Evaluate the improper integral or state that it is divergent. - ∫5∞dtt2−4t\int _ { 5 } ^ { \infty } \frac { d t } { t ^ { 2 } - 4 t }∫5∞​t2−4tdt​

Evaluate the improper integral. - ∫049dx49−x\int _ { 0 } ^ { 49 } \frac { d x } { \sqrt { 49 - x } }∫049​49−x​dx​

Evaluate the integral. - ∫π/12π/6cot⁡43tdt\int _ { \pi / 12 } ^ { \pi / 6 } \cot ^ { 4 } 3 t d t∫π/12π/6​cot43tdt

Integrate the function. - ∫dxx16x2−36\int \frac { d x } { x \sqrt { 16 x ^ { 2 } - 36 } }∫x16x2−36​dx​

Solve the problem. -Find the volume of the solid generated by revolving the region in the first quadrant bounded by y=exy = e ^ { x }y=ex and the xxx -axis, from x=0x = 0x=0 to x=ln⁡3x = \ln 3x=ln3 , about the yyy -axis.

Evaluate the integral. - ∫8xexdx\int 8 x e ^ { x } d x∫8xexdx

Evaluate the improper integral or state that it is divergent. - ∫−∞0dx(25+x)x\int _ { - \infty } ^ { 0 } \frac { d x } { ( 25 + x ) \sqrt { x } }∫−∞0​(25+x)x​dx​

Find the surface area or volume. -Use an integral table and a calculator to find to two decimal places the area of the surface generated by revolving the curve y=x2,−1≤x≤3y = x ^ { 2 } , - 1 \leq x \leq 3y=x2,−1≤x≤3 , about the xxx -axis.

Use Simpson's Rule with n = 4 steps to estimate the integral. - ∫02(x4+4)dx\int _ { 0 } ^ { 2 } \left( x ^ { 4 } + 4 \right) d x∫02​(x4+4)dx

Evaluate the improper integral. - ∫918/3dttt2−81\int _ { 9 } ^ { 18 / \sqrt { 3 } } \frac { d t } { t \sqrt { t ^ { 2 } - 81 } }∫918/3​​tt2−81​dt​

Find the surface area or volume. -The region between the curve y=sin⁡x,0≤x≤1.9y = \sin x , 0 \leq x \leq 1.9y=sinx,0≤x≤1.9 , and the xxx -axis is revolved about the xxx -axis to generate a solid. Use a table of integrals to find, to two decimal places, the volume of the solid generated.

Evaluate the integral. - ∫0π1−cos⁡2xdx\int _ { 0 } ^ { \pi } \sqrt { 1 - \cos ^ { 2 } x } d x∫0π​1−cos2x​dx

Evaluate the integral. - ∫xsin⁡−1xdx\int x \sin ^ { - 1 } x d x∫xsin−1xdx

Evaluate the improper integral or state that it is divergent. - ∫0∞2dx64+x2\int _ { 0 } ^ { \infty } \frac { 2 d x } { 64 + x ^ { 2 } }∫0∞​64+x22dx​

Use reduction formulas to evaluate the integral. - ∫3cos⁡3xsin⁡6xdx\int 3 \cos ^ { 3 } x \sin ^ { 6 } x d x∫3cos3xsin6xdx