Question 1

Integrate the function.
-$$\int \frac { d x } { \left( x ^ { 2 } + 25 \right) ^ { 3 / 2 } }$$

Accepted Answer

A)  $\frac { 5 } { x \sqrt { 25 - x ^ { 2 } } } + C$ 
B)  $\frac { x } { 5 \sqrt { 25 + x ^ { 2 } } } + C$ 
C)  $\frac { x } { 25 \sqrt { 25 - x ^ { 2 } } } + \frac { \sqrt { 25 - x ^ { 2 } } } { x } + C$ 
D)  $\frac { x } { 25 \sqrt { 25 + x ^ { 2 } } } + C$ 
A)  $\frac { 5 } { x \sqrt { 25 - x ^ { 2 } } } + C$ 
B)  $\frac { x } { 5 \sqrt { 25 + x ^ { 2 } } } + C$ 
C)  $\frac { x } { 25 \sqrt { 25 - x ^ { 2 } } } + \frac { \sqrt { 25 - x ^ { 2 } } } { x } + C$ 
D)  $\frac { x } { 25 \sqrt { 25 + x ^ { 2 } } } + C$

Question 2

Determine whether the improper integral converges or diverges.
-$$\int _ { 1 } ^ { \infty } \frac { \sqrt { x } } { \sqrt { x ^ { 4 } + 8 } } d x$$

Accepted Answer

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

Question 3

Solve the problem.
-Find the volume of the solid generated by revolving the region in the first quadrant bounded by the $x$-axis and the curve $y = \sin 6 x , 0 \leq x \leq \pi / 6$ about the line $x = \pi / 6$.

Accepted Answer

A)  $\frac { \pi ^ { 2 } } { 36 }$ 
B)  $\frac { 1 } { 18 } \pi ^ { 2 }$ 
C)  $\frac { 1 } { 18 } \pi$ 
D)  $\frac { 1 } { 18 } \pi ^ { 2 } - \pi$ 
A)  $\frac { \pi ^ { 2 } } { 36 }$ 
B)  $\frac { 1 } { 18 } \pi ^ { 2 }$ 
C)  $\frac { 1 } { 18 } \pi$ 
D)  $\frac { 1 } { 18 } \pi ^ { 2 } - \pi$

Question 4

Determine whether the improper integral converges or diverges.
-$$\int _ { 1 } ^ { \infty } \frac { x ^ { 6 } } { \sqrt { e ^ { x } - 1 } } d x$$

Accepted Answer

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

Question 5

Solve the problem by integration.
-The general expression for the slope of a curve is $\frac { 4 x + 4 } { x ^ { 2 } + 4 x }$. Find the equation of the curve if it passes through (1, 0 ).

Accepted Answer

A)  $y = \ln \left| \frac { x ( x - 4 ) ^ { 3 } } { 125 } \right|$ 
B)  $y = \ln \left| \frac { x ( x + 4 ) ^ { 3 } } { 5 } \right|$ 
C)  $y = \ln \left| x ( x + 4 ) ^ { 3 } \right|$ 
D)  $y = \ln \left| \frac { x ( x + 4 ) ^ { 3 } } { 125 } \right|$ 
A)  $y = \ln \left| \frac { x ( x - 4 ) ^ { 3 } } { 125 } \right|$ 
B)  $y = \ln \left| \frac { x ( x + 4 ) ^ { 3 } } { 5 } \right|$ 
C)  $y = \ln \left| x ( x + 4 ) ^ { 3 } \right|$ 
D)  $y = \ln \left| \frac { x ( x + 4 ) ^ { 3 } } { 125 } \right|$

Question 6

Integrate the function.
-$$\int \frac { 1 } { t ^ { 2 } \sqrt { 6 - t ^ { 2 } } } d t$$

Accepted Answer

A)  $- \frac { \sqrt { 6 - t ^ { 2 } } } { 6 t ^ { 2 } } + C$ 
B)  $- \frac { \sqrt { 6 - t ^ { 2 } } } { 6 t } + C$ 
C)  $- \frac { \sqrt { 6 - t ^ { 2 } } } { 6 t } + \sin ^ { - 1 } 6 t + C$ 
D)  $\frac { \sqrt { 6 - t ^ { 2 } } } { t } + C$ 
A)  $- \frac { \sqrt { 6 - t ^ { 2 } } } { 6 t ^ { 2 } } + C$ 
B)  $- \frac { \sqrt { 6 - t ^ { 2 } } } { 6 t } + C$ 
C)  $- \frac { \sqrt { 6 - t ^ { 2 } } } { 6 t } + \sin ^ { - 1 } 6 t + C$ 
D)  $\frac { \sqrt { 6 - t ^ { 2 } } } { t } + C$

Question 7

Evaluate the integral.
-$$\int 2 \sec ^ { 4 } x d x$$

Accepted Answer

A)  $2 \tan x + \frac { 2 } { 3 } \tan ^ { 3 } x + C$ 
B)  $- \frac { 2 } { 3 } \tan ^ { 3 } x + C$ 
C)  $2 ( \sec x + \tan x ) ^ { 5 } + C$ 
D)  $\frac { 2 } { 3 } \tan ^ { 3 } x + C$ 
A)  $2 \tan x + \frac { 2 } { 3 } \tan ^ { 3 } x + C$ 
B)  $- \frac { 2 } { 3 } \tan ^ { 3 } x + C$ 
C)  $2 ( \sec x + \tan x ) ^ { 5 } + C$ 
D)  $\frac { 2 } { 3 } \tan ^ { 3 } x + C$

Question 8

Solve the problem.
-Estimate the minimum number of subintervals needed to approximate the integral $\int _ { 1 } ^ { 4 } \left( 4 x ^ { 2 } + 7 \right) d x$ with an error of magnitude less than $10 ^ { - 4 }$ using the Trapezoidal Rule.

Accepted Answer

A)  213 
B)  849 
C)  425 
D)  82 
A)  213 
B)  849 
C)  425 
D)  82

Question 9

Determine whether the function is a probability density function over the given interval.
-$$\mathrm { f } ( \mathrm { x } ) = \frac { 1 } { 3 } \mathrm { x } , 6 \leq \mathrm { x } \leq 8$$

Accepted Answer

A) True 
 B)False

Question 10

Solve the problem by integration.
-$\text { Find the } x \text {-coordinate of the centroid of the first-quadrant area bounded by } y = \frac { 2 } { x ^ { 3 } + x } , x = 1 \text {, and } x = 3 \text {. }$

Accepted Answer

A)  0.789 
B)  -0.908 
C)  -1.815 
D)  1.578 
A)  0.789 
B)  -0.908 
C)  -1.815 
D)  1.578

Question 11

Solve the problem.
-Find the area bounded by $y = \frac { 3 } { \sqrt { 81 - 9 x ^ { 2 } } } , x = 0 , y = 0$, and $x = 2$.

Accepted Answer

A)  $\sin ^ { - 1 } \left( \frac { 2 } { 3 } \right)$ 
B)  $\frac { 1 } { 3 } \tan ^ { - 1 } \left( \frac { 2 } { 9 } \right)$ 
C)  $\frac { 1 } { 9 } \sin ^ { - 1 } \left( \frac { 2 } { 3 } \right)$ 
D)  $\frac { 1 } { 9 } \tan ^ { - 1 } \left( \frac { 2 } { 3 } \right)$ 
A)  $\sin ^ { - 1 } \left( \frac { 2 } { 3 } \right)$ 
B)  $\frac { 1 } { 3 } \tan ^ { - 1 } \left( \frac { 2 } { 9 } \right)$ 
C)  $\frac { 1 } { 9 } \sin ^ { - 1 } \left( \frac { 2 } { 3 } \right)$ 
D)  $\frac { 1 } { 9 } \tan ^ { - 1 } \left( \frac { 2 } { 3 } \right)$

Question 12

Find the surface area or volume.
-Use numerical integration with a programmable calculator or a CAS to find, to two decimal places, the area of the surface generated by revolving the curve $y = \sin 2 x , 0 \leq x \leq \frac { \pi } { 2 }$, about the $x$-axis.

Accepted Answer

A)  9.29 
B)  14.42 
C)  7.21 
D)  1.48 
A)  9.29 
B)  14.42 
C)  7.21 
D)  1.48

Question 13

Evaluate the improper integral or state that it is divergent.
-$$\int _ { - \infty } ^ { - 5 } \frac { 2 } { x ^ { 3 } } d x$$

Accepted Answer

A)  $- \frac { 1 } { 390625 }$ 
B)  $\frac { 2 } { 25 }$ 
C)  Divergent 
D)  $- \frac { 1 } { 25 }$ 
A)  $- \frac { 1 } { 390625 }$ 
B)  $\frac { 2 } { 25 }$ 
C)  Divergent 
D)  $- \frac { 1 } { 25 }$

Question 14

Evaluate the integral.
-$$\int - 6 x \cos 9 x d x$$

Accepted Answer

A)  $- \frac { 2 } { 3 } \cos 9 x - 6 x \sin 9 x + C$ 
B)  $- \frac { 2 } { 27 } \cos 9 x - \frac { 2 } { 3 } x \sin 9 x + C$ 
C)  $- \frac { 2 } { 27 } \cos 9 x - \frac { 2 } { 3 } x \sin 6 x + C$ 
D)  $- \frac { 2 } { 27 } \cos 9 x - \frac { 2 } { 3 } \sin 9 x + C$ 
A)  $- \frac { 2 } { 3 } \cos 9 x - 6 x \sin 9 x + C$ 
B)  $- \frac { 2 } { 27 } \cos 9 x - \frac { 2 } { 3 } x \sin 9 x + C$ 
C)  $- \frac { 2 } { 27 } \cos 9 x - \frac { 2 } { 3 } x \sin 6 x + C$ 
D)  $- \frac { 2 } { 27 } \cos 9 x - \frac { 2 } { 3 } \sin 9 x + C$

Question 15

Provide an appropriate response.
-Here is an argument that $\ln 2 = \infty - \infty$. Where does the argument go wrong?
$\ln 2 = \ln 1 + \ln 2 = \ln 1 - \ln \frac { 1 } { 2 }$
$= \lim _ { b \rightarrow \infty  } \ln \left( \frac { b - 1 } { b } \right) - \ln \frac { 1 } { 2 }$ $= \lim _ { b \rightarrow \infty  } \left[ \ln \frac { x - 1 } { x } \right] _ { 2 } ^ { b }$
$= \lim _ { b \rightarrow \infty  } [ \ln ( x - 1 ) - \ln x ] _ { 2 } ^ { b }$
$= \lim _ { b \rightarrow \infty  } \int _ { 2 } ^ { b } \left( \frac { 1 } { x - 1 } - \frac { 1 } { x } \right) d x$
$= \int _ { 2 } ^ { \infty } \left( \frac { 1 } { x - 1 } - \frac { 1 } { x } \right) d x$
Here is an argument that $\ln 2 = \infty - \infty .$
$\ln 2 = \ln 1 + \ln 2 = \ln 1 - \ln \frac { 1 } { 2 }$
$= \lim _ { b \rightarrow \infty } \ln \left( \frac { b - 1 } { b } \right) - \ln \frac { 1 } { 2 }$
$= \lim _ { b \rightarrow \infty  } \left[ \ln \frac { x - 1 } { x } \right] _ { 2 } ^ { b }$
$= \lim _ { b \rightarrow \infty  } [ \ln ( x - 1 ) - \ln x ] { } _ { 2 } ^ { b }$
$= \lim _ { b \rightarrow  \infty } \int _ { 2 } ^ { b } \left( \frac { 1 } { x - 1 } - \frac { 1 } { x } \right) d x$
$= \int _ { 2 } ^ { \infty } \left( \frac { 1 } { x - 1 } - \frac { 1 } { x } \right) d x$
$= \int _ { 2 } ^ { \infty } \frac { 1 } { x - 1 } d x - \int _ { 2 } ^ { \infty } \frac { 1 } { x } d x$
$= \lim _ { b \rightarrow \infty } [ \ln ( x - 1 ) ] \frac { b } { b } - \lim [ \ln x ]$
$= \infty - \infty$
$= \int _ { 2 } ^ { \infty } \frac { 1 } { x - 1 } d x - \int _ { 2 } ^ { \infty } \frac { 1 } { x } d x$
$= \lim _ { b \rightarrow \infty } [ \ln ( x - 1 ) ] _ { 2 } ^ { b } - \lim _ { b \rightarrow \infty } [ \ln x ] _ { 2 } ^ { b }$
$= \infty - \infty$

Accepted Answer

The answer of Provide an appropriate response.
-Here is an argument...

Question 16

Evaluate the improper integral or state that it is divergent.
-$$\int _ { - \infty } ^ { \infty } \frac { 10 x } { \left( x ^ { 2 } - 1 \right) ^ { 2 } } d x$$

Accepted Answer

A)  Divergent 
B)  10 
C)  20 
D)  0 
A)  Divergent 
B)  10 
C)  20 
D)  0

Question 17

Solve the problem.
-Find the length of the curve $y = \ln ( \sin x ) , \pi / 4 \leq x \leq \pi / 2$

Accepted Answer

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

Question 18

Solve the problem.
-Two dice are rolled, and the random variable X assigns to each outcome the sum of the number of dots showing on each face. What is the probability that X > 10?

Accepted Answer

A)  $\frac { 1 } { 18 }$ 
B)  $\frac { 5 } { 18 }$ 
C)  3 
D)  $\frac { 1 } { 12 }$ 
A)  $\frac { 1 } { 18 }$ 
B)  $\frac { 5 } { 18 }$ 
C)  3 
D)  $\frac { 1 } { 12 }$

Question 19

Integrate the function.
-$$\int \frac { d x } { \left( 16 x ^ { 2 } + 1 \right) ^ { 2 } }$$

Accepted Answer

A)  $\frac { 4 x } { 16 x ^ { 2 } + 1 } + C$ 
B)  $\ln \left| \sqrt { 16 x ^ { 2 } + 1 } + 4 x \right| + C$ 
C)  $\frac { 1 } { 8 } \tan ^ { - 1 } 4 x + \frac { x } { 32 x ^ { 2 } + 2 } + C$ 
D)  $\tan ^ { - 1 } 4 x - \frac { 4 x } { 16 x ^ { 2 } + 1 } + C$ 
A)  $\frac { 4 x } { 16 x ^ { 2 } + 1 } + C$ 
B)  $\ln \left| \sqrt { 16 x ^ { 2 } + 1 } + 4 x \right| + C$ 
C)  $\frac { 1 } { 8 } \tan ^ { - 1 } 4 x + \frac { x } { 32 x ^ { 2 } + 2 } + C$ 
D)  $\tan ^ { - 1 } 4 x - \frac { 4 x } { 16 x ^ { 2 } + 1 } + C$

Question 20

Solve the problem by integration.
-Under certain conditions, the velocity $\mathrm { v }$ (in $\mathrm { m } / \mathrm { s }$ ) of an object moving along a straight line as a function of the time $t$ (in s) is given by $v = \frac { 2 t ^ { 2 } + 20 t + 33 } { ( 4 t + 1 ) ( t + 4 ) ^ { 2 } }$. Find the distance traveled by the object during the first $2 \mathrm {~s}$.

Accepted Answer

A)  4.478 m 
B)  1.182 m 
C)  0.377 m 
D)  1.015 m 
A)  4.478 m 
B)  1.182 m 
C)  0.377 m 
D)  1.015 m

Integrate the function. - $\int \frac { d x } { \left( x ^ { 2 } + 25 \right) ^ { 3 / 2 } }$

Determine whether the improper integral converges or diverges. - $\int _ { 1 } ^ { \infty } \frac { \sqrt { x } } { \sqrt { x ^ { 4 } + 8 } } d x$

Solve the problem. -Find the volume of the solid generated by revolving the region in the first quadrant bounded by the $x$ -axis and the curve $y = \sin 6 x , 0 \leq x \leq \pi / 6$ about the line $x = \pi / 6$ .

Determine whether the improper integral converges or diverges. - $\int _ { 1 } ^ { \infty } \frac { x ^ { 6 } } { \sqrt { e ^ { x } - 1 } } d x$

Solve the problem by integration. -The general expression for the slope of a curve is $\frac { 4 x + 4 } { x ^ { 2 } + 4 x }$ . Find the equation of the curve if it passes through (1, 0 ).

Integrate the function. - $\int \frac { 1 } { t ^ { 2 } \sqrt { 6 - t ^ { 2 } } } d t$

Evaluate the integral. - $\int 2 \sec ^ { 4 } x d x$

Solve the problem. -Estimate the minimum number of subintervals needed to approximate the integral $\int _ { 1 } ^ { 4 } \left( 4 x ^ { 2 } + 7 \right) d x$ with an error of magnitude less than $10 ^ { - 4 }$ using the Trapezoidal Rule.

Determine whether the function is a probability density function over the given interval. - $\mathrm { f } ( \mathrm { x } ) = \frac { 1 } { 3 } \mathrm { x } , 6 \leq \mathrm { x } \leq 8$

Solve the problem by integration. - $\text { Find the } x \text {-coordinate of the centroid of the first-quadrant area bounded by } y = \frac { 2 } { x ^ { 3 } + x } , x = 1 \text {, and } x = 3 \text {. }$

Solve the problem. -Find the area bounded by $y = \frac { 3 } { \sqrt { 81 - 9 x ^ { 2 } } } , x = 0 , y = 0$ , and $x = 2$ .

Find the surface area or volume. -Use numerical integration with a programmable calculator or a CAS to find, to two decimal places, the area of the surface generated by revolving the curve $y = \sin 2 x , 0 \leq x \leq \frac { \pi } { 2 }$ , about the $x$ -axis.

Evaluate the improper integral or state that it is divergent. - $\int _ { - \infty } ^ { - 5 } \frac { 2 } { x ^ { 3 } } d x$

Evaluate the integral. - $\int - 6 x \cos 9 x d x$

Evaluate the improper integral or state that it is divergent. - $\int _ { - \infty } ^ { \infty } \frac { 10 x } { \left( x ^ { 2 } - 1 \right) ^ { 2 } } d x$

Solve the problem. -Find the length of the curve $y = \ln ( \sin x ) , \pi / 4 \leq x \leq \pi / 2$

Solve the problem. -Two dice are rolled, and the random variable X assigns to each outcome the sum of the number of dots showing on each face. What is the probability that X > 10?

Integrate the function. - $\int \frac { d x } { \left( 16 x ^ { 2 } + 1 \right) ^ { 2 } }$

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Exam 9: Techniques of Integration

Integrate the function. - ∫dx(x2+25)3/2\int \frac { d x } { \left( x ^ { 2 } + 25 \right) ^ { 3 / 2 } }∫(x2+25)3/2dx​

Determine whether the improper integral converges or diverges. - ∫1∞xx4+8dx\int _ { 1 } ^ { \infty } \frac { \sqrt { x } } { \sqrt { x ^ { 4 } + 8 } } d x∫1∞​x4+8​x​​dx

Solve the problem. -Find the volume of the solid generated by revolving the region in the first quadrant bounded by the xxx -axis and the curve y=sin⁡6x,0≤x≤π/6y = \sin 6 x , 0 \leq x \leq \pi / 6y=sin6x,0≤x≤π/6 about the line x=π/6x = \pi / 6x=π/6 .

Determine whether the improper integral converges or diverges. - ∫1∞x6ex−1dx\int _ { 1 } ^ { \infty } \frac { x ^ { 6 } } { \sqrt { e ^ { x } - 1 } } d x∫1∞​ex−1​x6​dx

Solve the problem by integration. -The general expression for the slope of a curve is 4x+4x2+4x\frac { 4 x + 4 } { x ^ { 2 } + 4 x }x2+4x4x+4​ . Find the equation of the curve if it passes through (1, 0 ).

Integrate the function. - ∫1t26−t2dt\int \frac { 1 } { t ^ { 2 } \sqrt { 6 - t ^ { 2 } } } d t∫t26−t2​1​dt

Evaluate the integral. - ∫2sec⁡4xdx\int 2 \sec ^ { 4 } x d x∫2sec4xdx

Solve the problem. -Estimate the minimum number of subintervals needed to approximate the integral ∫14(4x2+7)dx\int _ { 1 } ^ { 4 } \left( 4 x ^ { 2 } + 7 \right) d x∫14​(4x2+7)dx with an error of magnitude less than 10−410 ^ { - 4 }10−4 using the Trapezoidal Rule.

Determine whether the function is a probability density function over the given interval. - f(x)=13x,6≤x≤8\mathrm { f } ( \mathrm { x } ) = \frac { 1 } { 3 } \mathrm { x } , 6 \leq \mathrm { x } \leq 8f(x)=31​x,6≤x≤8

Solve the problem. -Find the area bounded by y=381−9x2,x=0,y=0y = \frac { 3 } { \sqrt { 81 - 9 x ^ { 2 } } } , x = 0 , y = 0y=81−9x2​3​,x=0,y=0 , and x=2x = 2x=2 .

Evaluate the improper integral or state that it is divergent. - ∫−∞−52x3dx\int _ { - \infty } ^ { - 5 } \frac { 2 } { x ^ { 3 } } d x∫−∞−5​x32​dx

Evaluate the integral. - ∫−6xcos⁡9xdx\int - 6 x \cos 9 x d x∫−6xcos9xdx

Evaluate the improper integral or state that it is divergent. - ∫−∞∞10x(x2−1)2dx\int _ { - \infty } ^ { \infty } \frac { 10 x } { \left( x ^ { 2 } - 1 \right) ^ { 2 } } d x∫−∞∞​(x2−1)210x​dx

Solve the problem. -Find the length of the curve y=ln⁡(sin⁡x),π/4≤x≤π/2y = \ln ( \sin x ) , \pi / 4 \leq x \leq \pi / 2y=ln(sinx),π/4≤x≤π/2