Question 1

Express the integrand as a sum of partial fractions and evaluate the integral.
-$$\int \frac { 3 x + 17 } { x ^ { 2 } + 9 x + 20 } d x$$

Accepted Answer

A)  $\ln \left| \frac { ( x + 4 ) ^ { 6 } } { ( x + 5 ) ^ { 5 } } \right| + C$ 
B)  $\ln \left| \frac { ( x + 4 ) ^ { 3 } } { ( x + 5 ) ^ { 5 } } \right| + C$ 
C)  $\ln \left| \frac { ( x + 4 ) ^ { 2 } } { ( x + 5 ) ^ { 5 } } \right| + C$ 
D)  $\ln \left| \frac { ( x + 4 ) ^ { 5 } } { ( x + 5 ) ^ { 2 } } \right| + C$ 
A)  $\ln \left| \frac { ( x + 4 ) ^ { 6 } } { ( x + 5 ) ^ { 5 } } \right| + C$ 
B)  $\ln \left| \frac { ( x + 4 ) ^ { 3 } } { ( x + 5 ) ^ { 5 } } \right| + C$ 
C)  $\ln \left| \frac { ( x + 4 ) ^ { 2 } } { ( x + 5 ) ^ { 5 } } \right| + C$ 
D)  $\ln \left| \frac { ( x + 4 ) ^ { 5 } } { ( x + 5 ) ^ { 2 } } \right| + C$

Question 2

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

Accepted Answer

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

Question 3

Solve the problem.
-The charge $\mathrm { q }$ (in coulombs) delivered by a current $\mathrm { i }$ (in amperes) is given by $\mathrm { q } = \int \mathrm { i } \mathrm { dt }$, where $t$ is the time (in seconds). A damped-out periodic wave form has current given by $\mathrm { i } = \mathrm { e } ^ { - 3 \mathrm { t } } \cos 5 \mathrm { t }$. Find a formula for the charge delivered over time $t$.

Accepted Answer

A)  $\frac { e ^ { - 3 t } ( - 3 \cos 5 t + 5 \sin 5 t ) } { 34 } + C$ 
B)  $\frac { e ^ { - 3 t } ( - 3 \cos 5 t + \sin 5 ) } { 34 } + C$ 
C)  $\frac { e ^ { - 3 t } ( - 3 \cos 5 t + 5 \sin 5 t ) } { 25 } + C$ 
D)  $\frac { - 3 \cos 5 t + 5 \sin 5 t } { 34 } + C$ 
A)  $\frac { e ^ { - 3 t } ( - 3 \cos 5 t + 5 \sin 5 t ) } { 34 } + C$ 
B)  $\frac { e ^ { - 3 t } ( - 3 \cos 5 t + \sin 5 ) } { 34 } + C$ 
C)  $\frac { e ^ { - 3 t } ( - 3 \cos 5 t + 5 \sin 5 t ) } { 25 } + C$ 
D)  $\frac { - 3 \cos 5 t + 5 \sin 5 t } { 34 } + C$

Question 4

Evaluate the integral.
-$$\int \cos \frac { \theta } { 2 } \cos \frac { \theta } { 5 } d \theta$$

Accepted Answer

A)  $\frac { 5 } { 3 } \sin \frac { 3 } { 10 } \theta + \frac { 5 } { 7 } \sin \frac { 7 } { 10 } \theta + C$ 
B)  $\frac { 5 } { 3 } \sin \frac { 3 } { 10 } \theta - \frac { 5 } { 7 } \sin \frac { 7 } { 10 } \theta + C$ 
C)  $\frac { 5 } { 3 } \cos \frac { 3 } { 10 } \theta + \frac { 5 } { 7 } \cos \frac { 7 } { 10 } \theta + C$ 
D)  $\frac { 5 } { 3 } \sin 3 \theta + \frac { 5 } { 7 } \sin 7 \theta + C$ 
A)  $\frac { 5 } { 3 } \sin \frac { 3 } { 10 } \theta + \frac { 5 } { 7 } \sin \frac { 7 } { 10 } \theta + C$ 
B)  $\frac { 5 } { 3 } \sin \frac { 3 } { 10 } \theta - \frac { 5 } { 7 } \sin \frac { 7 } { 10 } \theta + C$ 
C)  $\frac { 5 } { 3 } \cos \frac { 3 } { 10 } \theta + \frac { 5 } { 7 } \cos \frac { 7 } { 10 } \theta + C$ 
D)  $\frac { 5 } { 3 } \sin 3 \theta + \frac { 5 } { 7 } \sin 7 \theta + C$

Question 5

Integrate the function.
-$$\int \frac { \sqrt { 64 - x ^ { 2 } } } { x ^ { 4 } } d x , x < 8$$

Accepted Answer

A)  $- \frac { 192 x ^ { 3 } } { \left( 64 - x ^ { 2 } \right) ^ { 3 / 2 } } + C$ 
B)  $- \frac { \left( 64 - x ^ { 2 } \right) ^ { 3 / 2 } } { 192 x ^ { 3 } } + C$ 
C)  $\frac { 64 } { \left( 64 - x ^ { 2 } \right) ^ { 1 / 2 } } + C$ 
D)  $\frac { \left( 64 - x ^ { 2 } \right) ^ { 3 / 2 } } { x ^ { 3 } } + C$ 
A)  $- \frac { 192 x ^ { 3 } } { \left( 64 - x ^ { 2 } \right) ^ { 3 / 2 } } + C$ 
B)  $- \frac { \left( 64 - x ^ { 2 } \right) ^ { 3 / 2 } } { 192 x ^ { 3 } } + C$ 
C)  $\frac { 64 } { \left( 64 - x ^ { 2 } \right) ^ { 1 / 2 } } + C$ 
D)  $\frac { \left( 64 - x ^ { 2 } \right) ^ { 3 / 2 } } { x ^ { 3 } } + C$

Question 6

Solve the problem.
-Find an upper bound for $\left| E _ { S } \right|$ in estimating $\int _ { 0 } ^ { \pi } 5 x \cos x d x$ with $n = 12$ steps. Give your answer as a decimal rounded to five decimal places.

Accepted Answer

A)  0.00164 
B)  0.00172 
C)  0.00129 
D)  0.00205 
A)  0.00164 
B)  0.00172 
C)  0.00129 
D)  0.00205

Question 7

Solve the problem.
-Estimate the area of the surface generated by revolving the curve $y = 2 x ^ { 2 } , 0 \leq x \leq 3$ about the $x$-axis. Use Simpson's Rule with $\mathrm { n } = 6$.

Accepted Answer

A)  1007.254 
B)  1024.885 
C)  996.028 
D)  1021.107 
A)  1007.254 
B)  1024.885 
C)  996.028 
D)  1021.107

Question 8

Solve the problem.
-Estimate the minimum number of subintervals needed to approximate the integral $\int _ { - \pi / 2 } ^ { \pi / 2 } 4 \sin x d x$ with an error of magnitude less than $10 ^ { - 4 }$ using the Trapezoidal Rule.

Accepted Answer

A)  58 
B)  114 
C)  322 
D)  228 
A)  58 
B)  114 
C)  322 
D)  228

Question 9

Evaluate the improper integral or state that it is divergent.
-$$\int _ { 1 } ^ { \infty } \frac { 1 } { x \left( x ^ { 2 } + 5 \right) } d x$$

Accepted Answer

A)  $\ln 6$ 
B)  $\frac { \ln 6 } { 10 }$ 
C)  Divergent 
D)  $\ln 4$ 
A)  $\ln 6$ 
B)  $\frac { \ln 6 } { 10 }$ 
C)  Divergent 
D)  $\ln 4$

Question 10

Solve the problem.
-Estimate the minimum number of subintervals needed to approximate the integral $\int _ { 2 } ^ { 5 } \frac { 1 } { x - 1 } d x$ with an error of magnitude less than $10 ^ { - 4 }$ using Simpson's Rule.

Accepted Answer

A)  570 
B)  24 
C)  6 
D)  48 
A)  570 
B)  24 
C)  6 
D)  48

Question 11

Provide an appropriate response.
-this integral necessarily also diverge?

Accepted Answer

A) True 
 B)False

Question 12

Provide an appropriate response.
-$\text { Show that } \int _ { - \infty } ^ { 3 } \frac { \mathrm { dx } } { 1 + \mathrm { x } ^ { 2 } } + \int _ { 3 } ^ { \infty } \frac { \mathrm { dx } } { 1 + \mathrm { x } ^ { 2 } } = \int _ { - \infty } ^ { 5 } \frac { \mathrm { dx } } { 1 + \mathrm { x } ^ { 2 } } + \int _ { 5 } ^ { \infty } \frac { \mathrm { dx } } { 1 + \mathrm { x } ^ { 2 } }$

Accepted Answer

None

Question 13

Evaluate the integral.
-$$\int ( 3 x + 7 ) e ^ { - 5 x } d x$$

Accepted Answer

A)  $- \frac { 3 } { 5 } x e ^ { - 5 x } - \frac { 38 } { 25 } e ^ { - 5 x } + C$ 
B)  $- 15 x e ^ { - 5 x } - 110 e ^ { - 5 x } + C$ 
C)  $\frac { 3 } { 5 } x e ^ { - 5 x } + \frac { 38 } { 25 } e ^ { - 5 x } + C$ 
D)  $- \frac { 3 } { 5 } x e ^ { - 5 x } - e ^ { - 5 x } + C$ 
A)  $- \frac { 3 } { 5 } x e ^ { - 5 x } - \frac { 38 } { 25 } e ^ { - 5 x } + C$ 
B)  $- 15 x e ^ { - 5 x } - 110 e ^ { - 5 x } + C$ 
C)  $\frac { 3 } { 5 } x e ^ { - 5 x } + \frac { 38 } { 25 } e ^ { - 5 x } + C$ 
D)  $- \frac { 3 } { 5 } x e ^ { - 5 x } - e ^ { - 5 x } + C$

Question 14

Evaluate the integral.
-$$\int \frac { \sqrt { 4 + 3 x } } { x } d x$$

Accepted Answer

A)  $2 \sqrt { 4 + 3 x } + C$ 
B)  $2 \sqrt { 4 + 3 x } - 2 \ln \left( \frac { \sqrt { 4 + 3 x } - 2 } { \sqrt { 4 + 3 x } + 2 } \right) + C$ 
C)  $2 \sqrt { 4 + 3 x } + 2 \tan ^ { - 1 } \left( \frac { \sqrt { 4 + 3 x } } { 2 } \right) + C$ 
D)  $2 \sqrt { 4 + 3 x } + 2 \ln \left( \frac { \sqrt { 4 + 3 x } - 2 } { \sqrt { 4 + 3 x } + 2 } \right) + C$ 
A)  $2 \sqrt { 4 + 3 x } + C$ 
B)  $2 \sqrt { 4 + 3 x } - 2 \ln \left( \frac { \sqrt { 4 + 3 x } - 2 } { \sqrt { 4 + 3 x } + 2 } \right) + C$ 
C)  $2 \sqrt { 4 + 3 x } + 2 \tan ^ { - 1 } \left( \frac { \sqrt { 4 + 3 x } } { 2 } \right) + C$ 
D)  $2 \sqrt { 4 + 3 x } + 2 \ln \left( \frac { \sqrt { 4 + 3 x } - 2 } { \sqrt { 4 + 3 x } + 2 } \right) + C$

Question 15

Solve the problem.
-Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve $y = e ^ { x }$, and the line $x = \ln 5$ about the line $x = \ln 5$.

Accepted Answer

A)  $2 \pi ( 5 - \ln 5 )$ 
B)  $2 \pi ( 5 - \ln 6 )$ 
C)  $2 \pi ( 4 + \ln 5 )$ 
D)  $2 \pi ( 4 - \ln 5 )$ 
A)  $2 \pi ( 5 - \ln 5 )$ 
B)  $2 \pi ( 5 - \ln 6 )$ 
C)  $2 \pi ( 4 + \ln 5 )$ 
D)  $2 \pi ( 4 - \ln 5 )$

Question 16

Evaluate the integral by first performing long division on the integrand and then writing the proper fraction as a sum of partial fractions.
-$$\int \frac { 5 x ^ { 3 } + 7 x ^ { 2 } - 2 x - 4 } { x ^ { 3 } - x ^ { 2 } } d x$$

Accepted Answer

A)  $\frac { 5 } { x } + 6 \ln | x | - \frac { 4 } { x ^ { 2 } } + 6 \ln | x - 1 | + C$ 
B)  $5 x + 6 \ln \left| x ^ { 2 } \right| - \frac { 4 } { x ^ { 2 } } + 6 \ln | x - 1 | + C$ 
C)  $5 x + 7 \ln | x | + \frac { 4 } { x } + 6 \ln | x - 1 | + C$ 
D)  $5 x + 6 \ln | x | - \frac { 4 } { x } + 6 \ln | x - 1 | + C$ 
A)  $\frac { 5 } { x } + 6 \ln | x | - \frac { 4 } { x ^ { 2 } } + 6 \ln | x - 1 | + C$ 
B)  $5 x + 6 \ln \left| x ^ { 2 } \right| - \frac { 4 } { x ^ { 2 } } + 6 \ln | x - 1 | + C$ 
C)  $5 x + 7 \ln | x | + \frac { 4 } { x } + 6 \ln | x - 1 | + C$ 
D)  $5 x + 6 \ln | x | - \frac { 4 } { x } + 6 \ln | x - 1 | + C$

Question 17

Solve the problem by integration.
-Find the volume generated by revolving the first-quadrant area bounded by $y = \frac { x } { ( x + 2 ) ^ { 2 } }$ and $x = 2$ about the X-axis.

Accepted Answer

A)  $\frac { 1 } { 96 } \pi$ 
B)  $\frac { 1 } { 48 } \pi$ 
C)  $\frac { 1 } { 24 } \pi$ 
D)  $\frac { 1 } { 12 } \pi$ 
A)  $\frac { 1 } { 96 } \pi$ 
B)  $\frac { 1 } { 48 } \pi$ 
C)  $\frac { 1 } { 24 } \pi$ 
D)  $\frac { 1 } { 12 } \pi$

Question 18

Evaluate the integral.
-$$\int \frac { t ^ { 4 } \ln \left( t ^ { 5 } + 4 \right) } { t ^ { 5 } + 4 } d t$$

Accepted Answer

A)  $\frac { t ^ { 5 } \ln \left( t ^ { 5 } + 4 \right) } { t ^ { 5 } + 4 } + C$ 
B)  $\frac { \left[ \ln \left( t ^ { 5 } + 4 \right) \right] ^ { 2 } } { 10 } + C$ 
C)  $\frac { ( \ln t ) ^ { 2 } } { 10 } + C$ 
D)  $\left[ \ln \left( t ^ { 5 } + 4 \right) \right] ^ { 2 } + C$ 
A)  $\frac { t ^ { 5 } \ln \left( t ^ { 5 } + 4 \right) } { t ^ { 5 } + 4 } + C$ 
B)  $\frac { \left[ \ln \left( t ^ { 5 } + 4 \right) \right] ^ { 2 } } { 10 } + C$ 
C)  $\frac { ( \ln t ) ^ { 2 } } { 10 } + C$ 
D)  $\left[ \ln \left( t ^ { 5 } + 4 \right) \right] ^ { 2 } + C$

Question 19

Solve the problem.
-Find an upper bound for $\left| \mathrm { E } _ { \mathrm { T } } \right|$ in estimating $\int _ { 4 } ^ { 6 } \frac { 1 } { ( \mathrm { x } - 1 ) ^ { 2 } } \mathrm { dx }$ with $\mathrm { n } = 7$ steps.

Accepted Answer

A)  $\frac { 4 } { 30,625 }$ 
B)  $\frac { 4 } { 11,907 }$ 
C)  $\frac { 4 } { 3969 }$ 
D)  $\frac { 4 } { 147 }$ 
A)  $\frac { 4 } { 30,625 }$ 
B)  $\frac { 4 } { 11,907 }$ 
C)  $\frac { 4 } { 3969 }$ 
D)  $\frac { 4 } { 147 }$

Question 20

Solve the problem.
-Find an upper bound for $| E_ S |$ in estimating $\int _ { 1 } ^ { 3 } \left( 2 x ^ { 4 } - 2 x \right) d x$ with $n = 6$ steps.

Accepted Answer

A)  $\frac { 32 } { 81 }$ 
B)  $\frac { 8 } { 1215 }$ 
C)  $\frac { 4 } { 1215 }$ 
D)  1 
A)  $\frac { 32 } { 81 }$ 
B)  $\frac { 8 } { 1215 }$ 
C)  $\frac { 4 } { 1215 }$ 
D)  1

Express the integrand as a sum of partial fractions and evaluate the integral. - $\int \frac { 3 x + 17 } { x ^ { 2 } + 9 x + 20 } d x$

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

Evaluate the integral. - $\int \cos \frac { \theta } { 2 } \cos \frac { \theta } { 5 } d \theta$

Integrate the function. - $\int \frac { \sqrt { 64 - x ^ { 2 } } } { x ^ { 4 } } d x , x < 8$

Solve the problem. -Find an upper bound for $\left| E _ { S } \right|$ in estimating $\int _ { 0 } ^ { \pi } 5 x \cos x d x$ with $n = 12$ steps. Give your answer as a decimal rounded to five decimal places.

Solve the problem. -Estimate the area of the surface generated by revolving the curve $y = 2 x ^ { 2 } , 0 \leq x \leq 3$ about the $x$ -axis. Use Simpson's Rule with $\mathrm { n } = 6$ .

Solve the problem. -Estimate the minimum number of subintervals needed to approximate the integral $\int _ { - \pi / 2 } ^ { \pi / 2 } 4 \sin x d x$ with an error of magnitude less than $10 ^ { - 4 }$ using the Trapezoidal Rule.

Evaluate the improper integral or state that it is divergent. - $\int _ { 1 } ^ { \infty } \frac { 1 } { x \left( x ^ { 2 } + 5 \right) } d x$

Solve the problem. -Estimate the minimum number of subintervals needed to approximate the integral $\int _ { 2 } ^ { 5 } \frac { 1 } { x - 1 } d x$ with an error of magnitude less than $10 ^ { - 4 }$ using Simpson's Rule.

Provide an appropriate response. -this integral necessarily also diverge?

Evaluate the integral. - $\int ( 3 x + 7 ) e ^ { - 5 x } d x$

Evaluate the integral. - $\int \frac { \sqrt { 4 + 3 x } } { x } d x$

Solve the problem. -Find the volume of the solid generated by revolving the region in the first quadrant bounded by the coordinate axes, the curve $y = e ^ { x }$ , and the line $x = \ln 5$ about the line $x = \ln 5$ .

Evaluate the integral by first performing long division on the integrand and then writing the proper fraction as a sum of partial fractions. - $\int \frac { 5 x ^ { 3 } + 7 x ^ { 2 } - 2 x - 4 } { x ^ { 3 } - x ^ { 2 } } d x$

Solve the problem by integration. -Find the volume generated by revolving the first-quadrant area bounded by $y = \frac { x } { ( x + 2 ) ^ { 2 } }$ and $x = 2$ about the X-axis.

Evaluate the integral. - $\int \frac { t ^ { 4 } \ln \left( t ^ { 5 } + 4 \right) } { t ^ { 5 } + 4 } d t$

Solve the problem. -Find an upper bound for $\left| \mathrm { E } _ { \mathrm { T } } \right|$ in estimating $\int _ { 4 } ^ { 6 } \frac { 1 } { ( \mathrm { x } - 1 ) ^ { 2 } } \mathrm { dx }$ with $\mathrm { n } = 7$ steps.

Solve the problem. -Find an upper bound for $| E_ S |$ in estimating $\int _ { 1 } ^ { 3 } \left( 2 x ^ { 4 } - 2 x \right) d x$ with $n = 6$ steps.

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Exam 9: Techniques of Integration

Express the integrand as a sum of partial fractions and evaluate the integral. - ∫3x+17x2+9x+20dx\int \frac { 3 x + 17 } { x ^ { 2 } + 9 x + 20 } d x∫x2+9x+203x+17​dx

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

Evaluate the integral. - ∫cos⁡θ2cos⁡θ5dθ\int \cos \frac { \theta } { 2 } \cos \frac { \theta } { 5 } d \theta∫cos2θ​cos5θ​dθ

Integrate the function. - ∫64−x2x4dx,x<8\int \frac { \sqrt { 64 - x ^ { 2 } } } { x ^ { 4 } } d x , x < 8∫x464−x2​​dx,x<8

Solve the problem. -Find an upper bound for ∣ES∣\left| E _ { S } \right|∣ES​∣ in estimating ∫0π5xcos⁡xdx\int _ { 0 } ^ { \pi } 5 x \cos x d x∫0π​5xcosxdx with n=12n = 12n=12 steps. Give your answer as a decimal rounded to five decimal places.

Solve the problem. -Estimate the area of the surface generated by revolving the curve y=2x2,0≤x≤3y = 2 x ^ { 2 } , 0 \leq x \leq 3y=2x2,0≤x≤3 about the xxx -axis. Use Simpson's Rule with n=6\mathrm { n } = 6n=6 .

Solve the problem. -Estimate the minimum number of subintervals needed to approximate the integral ∫−π/2π/24sin⁡xdx\int _ { - \pi / 2 } ^ { \pi / 2 } 4 \sin x d x∫−π/2π/2​4sinxdx with an error of magnitude less than 10−410 ^ { - 4 }10−4 using the Trapezoidal Rule.

Evaluate the improper integral or state that it is divergent. - ∫1∞1x(x2+5)dx\int _ { 1 } ^ { \infty } \frac { 1 } { x \left( x ^ { 2 } + 5 \right) } d x∫1∞​x(x2+5)1​dx

Solve the problem. -Estimate the minimum number of subintervals needed to approximate the integral ∫251x−1dx\int _ { 2 } ^ { 5 } \frac { 1 } { x - 1 } d x∫25​x−11​dx with an error of magnitude less than 10−410 ^ { - 4 }10−4 using Simpson's Rule.