Question 1

Integrate the function.
-$$\int \frac { 28 d x } { x ^ { 2 } \sqrt { x ^ { 2 } + 16 } }$$

Accepted Answer

A)  $\frac { 7 \sqrt { x ^ { 2 } + 16 } } { 4 x } + C$ 
B)  $- \frac { 7 \sqrt { x ^ { 2 } + 16 } } { 4 x } + C$ 
C)  $\frac { 4 \sqrt { x ^ { 2 } + 16 } } { 7 x } + C$ 
D)  $- \frac { \sqrt { x ^ { 2 } + 16 } } { 7 x } + C$ 
A)  $\frac { 7 \sqrt { x ^ { 2 } + 16 } } { 4 x } + C$ 
B)  $- \frac { 7 \sqrt { x ^ { 2 } + 16 } } { 4 x } + C$ 
C)  $\frac { 4 \sqrt { x ^ { 2 } + 16 } } { 7 x } + C$ 
D)  $- \frac { \sqrt { x ^ { 2 } + 16 } } { 7 x } + C$

Question 2

Determine whether the improper integral converges or diverges.
-$$\int _ { 0 } ^ { 2 } \frac { d x } { | x - 1 | } d x$$

Accepted Answer

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

Question 3

Solve the problem by integration.
-The current $\mathrm { i }$ (in A) as a function of the time $t$ (in s) in a certain electric circuit is given by $i = \frac { 8 t + 4 } { 4 t ^ { 2 } + 4 t + 1 }$. Find the total charge that passes a given point in the circuit during the first second.

Accepted Answer

A)  0.549 C 
B)  8.789 C 
C)  2.079 C 
D)  2.197 C 
A)  0.549 C 
B)  8.789 C 
C)  2.079 C 
D)  2.197 C

Question 4

Evaluate the integral.
-Use the formula $\int f ^ { - 1 } ( x ) d x = x f ^ { - 1 } ( x ) - \int f ( y ) d y , y = f ^ { - 1 } ( x )$ to evaluate the integral. $\int \cot ^ { - 1 } x d x$

Accepted Answer

A)  $x \cot ^ { - 1 } x - \ln \sin x + C$ 
B)  $x \cot ^ { - 1 } x + \ln \sin \left( \cot ^ { - 1 } x \right) + C$ 
C)  $x \cot ^ { - 1 } x - \ln \sin \left( \cot ^ { - 1 } x \right) + C$ 
D)  $x \cot ^ { - 1 } x - x + C$ 
A)  $x \cot ^ { - 1 } x - \ln \sin x + C$ 
B)  $x \cot ^ { - 1 } x + \ln \sin \left( \cot ^ { - 1 } x \right) + C$ 
C)  $x \cot ^ { - 1 } x - \ln \sin \left( \cot ^ { - 1 } x \right) + C$ 
D)  $x \cot ^ { - 1 } x - x + C$

Question 5

Use the Trapezoidal Rule with n = 4 steps to estimate the integral.
-$$\int _ { 1 } ^ { 3 } ( 6 x + 4 ) d x$$

Accepted Answer

A)  20 
B)  32 
C)  64 
D)  16 
A)  20 
B)  32 
C)  64 
D)  16

Question 6

Solve the problem.
-There are 3 balls in a hat; one with the number 3 on it, one with the number 4 on it, and one with the number 7 on it. You pick a ball from the hat at random and then you flip a coin to obtain heads (H) or tails (T). Determine
The set of possible outcomes.

Accepted Answer

A)  {3 H, 4 H, 7 H} 
B)  {3 4 7 H, 3 4 7 T} 
C)  {3 H T, 4 H T, 7 H T} 
D)  {3 H, 3 T, 4 H, 4 T, 7 H, 7 T} 
A)  {3 H, 4 H, 7 H} 
B)  {3 4 7 H, 3 4 7 T} 
C)  {3 H T, 4 H T, 7 H T} 
D)  {3 H, 3 T, 4 H, 4 T, 7 H, 7 T}

Question 7

Solve the problem.
-The length of one arch of the curve $y = 3 \sin 2 x$ is given by
$$L = \int _ { 0 } ^ { \pi / 2 } \sqrt { 1 + 36 \cos ^ { 2 } 2 x } d x$$
Estimate $\mathrm { L }$ by the Trapezoidal Rule with $\mathrm { n } = 6$.

Accepted Answer

A)  6.2807 
B)  6.4115 
C)  5.8169 
D)  6.1995 
A)  6.2807 
B)  6.4115 
C)  5.8169 
D)  6.1995

Question 8

Integrate the function.
-$$\int \frac { d x } { \left( x ^ { 2 } - 64 \right) ^ { 3 / 2 } } , x > 8$$

Accepted Answer

A)  $\frac { \sqrt { x ^ { 2 } - 64 } } { x } + C$ 
B)  $\frac { 64 x } { \sqrt { x ^ { 2 } - 64 } } + C$ 
C)  $- \frac { x } { \sqrt { x ^ { 2 } - 64 } } + C$ 
D)  $- \frac { x } { 64 \sqrt { x ^ { 2 } - 64 } } + C$ 
A)  $\frac { \sqrt { x ^ { 2 } - 64 } } { x } + C$ 
B)  $\frac { 64 x } { \sqrt { x ^ { 2 } - 64 } } + C$ 
C)  $- \frac { x } { \sqrt { x ^ { 2 } - 64 } } + C$ 
D)  $- \frac { x } { 64 \sqrt { x ^ { 2 } - 64 } } + C$

Question 9

Provide an appropriate response.
-The error formula for the Trapezoidal Rule depends upon
i) $f ( x )$.
ii) $f ^ { \prime } ( x )$.
iii) $f ^ { \prime \prime } ( x )$.
iv) the number of steps

Accepted Answer

A)  iii and iv 
B)  $\mathrm { i }$ and $\mathrm { iii }$ 
C)  ii and iv 
D)  $\mathrm { i }$, iii, and iv 
A)  iii and iv 
B)  $\mathrm { i }$ and $\mathrm { iii }$ 
C)  ii and iv 
D)  $\mathrm { i }$, iii, and iv

Question 10

Solve the problem by integration.
-Under specified conditions, the time $t$ (in min) required to form $x$ grams of a substance during a chemical reaction is given by $t = \int \frac { d x } { ( 7 - x ) ( 2 - x ) }$. Find the equation relating $t$ and $x$ if $x = 0 g$ when $t = 0$ min.

Accepted Answer

A)  $t = \frac { 1 } { 5 } \ln \left| \frac { 2 - x } { 7 - x } \right| - \frac { 1 } { 5 } \ln \frac { 2 } { 7 }$ 
B)  $t = \frac { 1 } { 5 } \ln \left| \frac { 7 - x } { 2 - x } \right| + \frac { 1 } { 5 } \ln \frac { 7 } { 2 }$ 
C)  $t = \frac { 1 } { 5 } \ln \left| \frac { 7 - x } { 2 - x } \right| - \frac { 1 } { 5 } \ln \frac { 7 } { 2 }$ 
D)  $t = \frac { 1 } { 5 } \ln \left| \frac { 2 - x } { 7 - x } \right| + \frac { 1 } { 5 } \ln \frac { 2 } { 7 }$ 
A)  $t = \frac { 1 } { 5 } \ln \left| \frac { 2 - x } { 7 - x } \right| - \frac { 1 } { 5 } \ln \frac { 2 } { 7 }$ 
B)  $t = \frac { 1 } { 5 } \ln \left| \frac { 7 - x } { 2 - x } \right| + \frac { 1 } { 5 } \ln \frac { 7 } { 2 }$ 
C)  $t = \frac { 1 } { 5 } \ln \left| \frac { 7 - x } { 2 - x } \right| - \frac { 1 } { 5 } \ln \frac { 7 } { 2 }$ 
D)  $t = \frac { 1 } { 5 } \ln \left| \frac { 2 - x } { 7 - x } \right| + \frac { 1 } { 5 } \ln \frac { 2 } { 7 }$

Question 11

Evaluate the integral by first performing long division on the integrand and then writing the proper fraction as a sum of partial fractions.
-$$\int \frac { 7 x ^ { 3 } + 7 x ^ { 2 } + 2 } { x ^ { 2 } + x } d x$$

Accepted Answer

A)  $\frac { 7 } { 2 } x ^ { 2 } + 2 \ln | x - 1 | - 2 \ln | x | + C$ 
B)  $7 x ^ { 2 } + 2 \ln | x + 1 | - 2 \ln | x | + C$ 
C)  $2 \ln | x + 1 | - 2 \ln | x | + C$ 
D)  $\frac { 7 } { 2 } x ^ { 2 } - 2 \ln | x + 1 | + 2 \ln | x | + C$ 
A)  $\frac { 7 } { 2 } x ^ { 2 } + 2 \ln | x - 1 | - 2 \ln | x | + C$ 
B)  $7 x ^ { 2 } + 2 \ln | x + 1 | - 2 \ln | x | + C$ 
C)  $2 \ln | x + 1 | - 2 \ln | x | + C$ 
D)  $\frac { 7 } { 2 } x ^ { 2 } - 2 \ln | x + 1 | + 2 \ln | x | + C$

Question 12

Use a trigonometric substitution to evaluate the integral.
-$$\int \frac { d x } { x \sqrt { x ^ { 2 } - 9 } }$$

Accepted Answer

A)  $3 \sec ^ { - 1 } x + C$ 
B)  $\frac { 1 } { 3 } \sin ^ { - 1 } \frac { x } { 3 } + C$ 
C)  $\frac { 1 } { 3 } \sec ^ { - 1 } \left| \frac { x } { 3 } \right| + C$ 
D)  $\sin ^ { - 1 } \frac { x } { 3 } + C$ 
A)  $3 \sec ^ { - 1 } x + C$ 
B)  $\frac { 1 } { 3 } \sin ^ { - 1 } \frac { x } { 3 } + C$ 
C)  $\frac { 1 } { 3 } \sec ^ { - 1 } \left| \frac { x } { 3 } \right| + C$ 
D)  $\sin ^ { - 1 } \frac { x } { 3 } + C$

Question 13

Evaluate the integral by making a substitution (possibly trigonometric) and then applying a reduction formula.
-$$\int \frac { \sec ^ { 3 } \sqrt { \theta } } { \sqrt { \theta } } \mathrm { d } \theta$$

Accepted Answer

A)  $\sec \sqrt { \theta } \tan \sqrt { \theta } - \ln | \sec \sqrt { \theta } + \tan \sqrt { \theta } | + C$ 
B)  $- \sec \sqrt { \theta } \tan \sqrt { \theta } + \ln | \sec \sqrt { \theta } + \tan \sqrt { \theta } | + C$ 
C)  $\sec \sqrt { \theta } \tan \sqrt { \theta } + \ln | \sec \sqrt { \theta } + \tan \sqrt { \theta } | + C$ 
D)  $- \sec \sqrt { \theta } \tan \sqrt { \theta } + \ln | \sec \sqrt { \theta } - \tan \sqrt { \theta } | + C$ 
A)  $\sec \sqrt { \theta } \tan \sqrt { \theta } - \ln | \sec \sqrt { \theta } + \tan \sqrt { \theta } | + C$ 
B)  $- \sec \sqrt { \theta } \tan \sqrt { \theta } + \ln | \sec \sqrt { \theta } + \tan \sqrt { \theta } | + C$ 
C)  $\sec \sqrt { \theta } \tan \sqrt { \theta } + \ln | \sec \sqrt { \theta } + \tan \sqrt { \theta } | + C$ 
D)  $- \sec \sqrt { \theta } \tan \sqrt { \theta } + \ln | \sec \sqrt { \theta } - \tan \sqrt { \theta } | + C$

Question 14

Use a trigonometric substitution to evaluate the integral.
-$$\int _ { 0 } ^ { \ln 3 } \frac { e ^ { t } d t } { 25 + e ^ { 2 t } }$$

Accepted Answer

A)  -0.069 
B)  0.069 
C)  0.343 
D)  0.043 
A)  -0.069 
B)  0.069 
C)  0.343 
D)  0.043

Question 15

Solve the problem.
-Find an upper bound for $\left| E _ { S } \right|$ in estimating $\int _ { 2 } ^ { 4 } \frac { 1 } { x - 1 } d x$ with $n = 8$ steps.

Accepted Answer

A)  $\frac { 1 } { 960 }$ 
B)  $\frac { 1 } { 1920 }$ 
C)  $\frac { 1 } { 466,560 }$ 
D)  $\frac { 1 } { 30 }$ 
A)  $\frac { 1 } { 960 }$ 
B)  $\frac { 1 } { 1920 }$ 
C)  $\frac { 1 } { 466,560 }$ 
D)  $\frac { 1 } { 30 }$

Question 16

Evaluate the improper integral.
-$$\int _ { 0 } ^ { 9 } \frac { d x } { \sqrt { 81 - x ^ { 2 } } }$$

Accepted Answer

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

Question 17

Expand the quotient by partial fractions.
-$\frac { 6 z } { z ^ { 3 } - 2 z ^ { 2 } - 8 z }$

Accepted Answer

A)  $\frac { 1 } { z } + \frac { - 1 } { z - 4 } + \frac { 1 } { z + 2 }$ 
B)  $\frac { 1 } { z - 4 } + \frac { 1 } { z + 2 }$ 
C)  $\frac { 1 } { \mathrm { z } } + \frac { 1 } { \mathrm { z } - 4 } + \frac { - 1 } { \mathrm { z } + 2 }$ 
D)  $\frac { 1 } { z - 4 } + \frac { - 1 } { z + 2 }$ 
A)  $\frac { 1 } { z } + \frac { - 1 } { z - 4 } + \frac { 1 } { z + 2 }$ 
B)  $\frac { 1 } { z - 4 } + \frac { 1 } { z + 2 }$ 
C)  $\frac { 1 } { \mathrm { z } } + \frac { 1 } { \mathrm { z } - 4 } + \frac { - 1 } { \mathrm { z } + 2 }$ 
D)  $\frac { 1 } { z - 4 } + \frac { - 1 } { z + 2 }$

Question 18

Evaluate the improper integral or state that it is divergent.
-$$\int _ { 0 } ^ { \infty } \frac { 36 \left( 1 + \tan ^ { - 1 } x \right) } { 1 + x ^ { 2 } } d x$$

Accepted Answer

A)  $18 \pi$ 
B)  $18 \left( 1 + \frac { \pi } { 2 } \right) ^ { 2 }$ 
C)  $36 \ln \left( 1 + \frac { \pi } { 2 } \right)$ 
D)  $18 \pi \left( 1 + \frac { \pi } { 4 } \right)$ 
A)  $18 \pi$ 
B)  $18 \left( 1 + \frac { \pi } { 2 } \right) ^ { 2 }$ 
C)  $36 \ln \left( 1 + \frac { \pi } { 2 } \right)$ 
D)  $18 \pi \left( 1 + \frac { \pi } { 4 } \right)$

Question 19

Find the area or volume.
-Find the volume of the solid generated by revolving the area under $y = 8 \mathrm { e } ^ { - \mathrm { x } }$ in the first quadrant about the $\mathrm { x }$-axis

Accepted Answer

A)  $8 \pi$ 
B)  $128 \pi$ 
C)  $32 \pi$ 
D)  $64 \pi$ 
A)  $8 \pi$ 
B)  $128 \pi$ 
C)  $32 \pi$ 
D)  $64 \pi$

Question 20

Solve the problem.
-Find the area of the region enclosed by $y = 5 x \sin x$ and the $x$-axis for $0 \leq x \leq \pi$.

Accepted Answer

A)  $\pi$ 
B)  $\frac { 5 } { 2 } \pi$ 
C)  $10 \pi$ 
D)  $5 \pi$ 
A)  $\pi$ 
B)  $\frac { 5 } { 2 } \pi$ 
C)  $10 \pi$ 
D)  $5 \pi$

Integrate the function. - $\int \frac { 28 d x } { x ^ { 2 } \sqrt { x ^ { 2 } + 16 } }$

Determine whether the improper integral converges or diverges. - $\int _ { 0 } ^ { 2 } \frac { d x } { | x - 1 | } d x$

Solve the problem by integration. -The current $\mathrm { i }$ (in A) as a function of the time $t$ (in s) in a certain electric circuit is given by $i = \frac { 8 t + 4 } { 4 t ^ { 2 } + 4 t + 1 }$ . Find the total charge that passes a given point in the circuit during the first second.

Evaluate the integral. -Use the formula $\int f ^ { - 1 } ( x ) d x = x f ^ { - 1 } ( x ) - \int f ( y ) d y , y = f ^ { - 1 } ( x )$ to evaluate the integral. $\int \cot ^ { - 1 } x d x$

Use the Trapezoidal Rule with n = 4 steps to estimate the integral. - $\int _ { 1 } ^ { 3 } ( 6 x + 4 ) d x$

Solve the problem. -There are 3 balls in a hat; one with the number 3 on it, one with the number 4 on it, and one with the number 7 on it. You pick a ball from the hat at random and then you flip a coin to obtain heads (H) or tails (T). Determine The set of possible outcomes.

Solve the problem. -The length of one arch of the curve $y = 3 \sin 2 x$ is given by $L = \int _ { 0 } ^ { \pi / 2 } \sqrt { 1 + 36 \cos ^ { 2 } 2 x } d x$ Estimate $\mathrm { L }$ by the Trapezoidal Rule with $\mathrm { n } = 6$ .

Integrate the function. - $\int \frac { d x } { \left( x ^ { 2 } - 64 \right) ^ { 3 / 2 } } , x > 8$

Provide an appropriate response. -The error formula for the Trapezoidal Rule depends upon i) $f ( x )$ . ii) $f ^ { \prime } ( x )$ . iii) $f ^ { \prime \prime } ( x )$ . iv) the number of steps

Solve the problem by integration. -Under specified conditions, the time $t$ (in min) required to form $x$ grams of a substance during a chemical reaction is given by $t = \int \frac { d x } { ( 7 - x ) ( 2 - x ) }$ . Find the equation relating $t$ and $x$ if $x = 0 g$ when $t = 0$ min.

Evaluate the integral by first performing long division on the integrand and then writing the proper fraction as a sum of partial fractions. - $\int \frac { 7 x ^ { 3 } + 7 x ^ { 2 } + 2 } { x ^ { 2 } + x } d x$

Use a trigonometric substitution to evaluate the integral. - $\int \frac { d x } { x \sqrt { x ^ { 2 } - 9 } }$

Evaluate the integral by making a substitution (possibly trigonometric) and then applying a reduction formula. - $\int \frac { \sec ^ { 3 } \sqrt { \theta } } { \sqrt { \theta } } \mathrm { d } \theta$

Use a trigonometric substitution to evaluate the integral. - $\int _ { 0 } ^ { \ln 3 } \frac { e ^ { t } d t } { 25 + e ^ { 2 t } }$

Solve the problem. -Find an upper bound for $\left| E _ { S } \right|$ in estimating $\int _ { 2 } ^ { 4 } \frac { 1 } { x - 1 } d x$ with $n = 8$ steps.

Evaluate the improper integral. - $\int _ { 0 } ^ { 9 } \frac { d x } { \sqrt { 81 - x ^ { 2 } } }$

Expand the quotient by partial fractions. - $\frac { 6 z } { z ^ { 3 } - 2 z ^ { 2 } - 8 z }$

Evaluate the improper integral or state that it is divergent. - $\int _ { 0 } ^ { \infty } \frac { 36 \left( 1 + \tan ^ { - 1 } x \right) } { 1 + x ^ { 2 } } d x$

Find the area or volume. -Find the volume of the solid generated by revolving the area under $y = 8 \mathrm { e } ^ { - \mathrm { x } }$ in the first quadrant about the $\mathrm { x }$ -axis

Solve the problem. -Find the area of the region enclosed by $y = 5 x \sin x$ and the $x$ -axis for $0 \leq x \leq \pi$ .

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Exam 9: Techniques of Integration

Integrate the function. - ∫28dxx2x2+16\int \frac { 28 d x } { x ^ { 2 } \sqrt { x ^ { 2 } + 16 } }∫x2x2+16​28dx​

Determine whether the improper integral converges or diverges. - ∫02dx∣x−1∣dx\int _ { 0 } ^ { 2 } \frac { d x } { | x - 1 | } d x∫02​∣x−1∣dx​dx

Use the Trapezoidal Rule with n = 4 steps to estimate the integral. - ∫13(6x+4)dx\int _ { 1 } ^ { 3 } ( 6 x + 4 ) d x∫13​(6x+4)dx