Question 1

Expand the quotient by partial fractions.
-$\frac { y + 2 } { y ^ { 2 } ( y + 1 ) }$

Accepted Answer

A)  $\frac { - 1 } { y } + \frac { 2 } { y ^ { 2 } } + \frac { 1 } { y + 1 }$ 
B)  $\frac { - 1 } { y } + \frac { 2 } { y ^ { 2 } } + \frac { 2 } { y + 1 }$ 
C)  $\frac { 1 } { y } + \frac { 2 } { y ^ { 2 } } + \frac { 1 } { y + 1 }$ 
D)  $\frac { 2 } { y ^ { 2 } } + \frac { 1 } { y + 1 }$ 
A)  $\frac { - 1 } { y } + \frac { 2 } { y ^ { 2 } } + \frac { 1 } { y + 1 }$ 
B)  $\frac { - 1 } { y } + \frac { 2 } { y ^ { 2 } } + \frac { 2 } { y + 1 }$ 
C)  $\frac { 1 } { y } + \frac { 2 } { y ^ { 2 } } + \frac { 1 } { y + 1 }$ 
D)  $\frac { 2 } { y ^ { 2 } } + \frac { 1 } { y + 1 }$

Question 2

Integrate the function.
-$$\int _ { 0 } ^ { 1 } \frac { d x } { \sqrt { 81 - x ^ { 2 } } }$$

Accepted Answer

A)  $\frac { 1 } { 9 } \sin ^ { - 1 } \frac { 1 } { 9 }$ 
B)  $\cos ^ { - 1 } \frac { 1 } { 9 }$ 
C)  $9 \cos ^ { - 1 } \frac { 1 } { 9 }$ 
D)  $\sin ^ { - 1 } \frac { 1 } { 9 }$ 
A)  $\frac { 1 } { 9 } \sin ^ { - 1 } \frac { 1 } { 9 }$ 
B)  $\cos ^ { - 1 } \frac { 1 } { 9 }$ 
C)  $9 \cos ^ { - 1 } \frac { 1 } { 9 }$ 
D)  $\sin ^ { - 1 } \frac { 1 } { 9 }$

Question 3

For the given probability density function, over the stated interval, find the requested value.
-$f ( x ) = \frac { 1 } { 7 } x$, over $[ 1,4 ] ;$ Find the mean.

Accepted Answer

A)  $\frac { 6 } { 7 }$ 
B)  $\frac { 62 } { 21 }$ 
C)  $\frac { 64 } { 21 }$ 
D)  3 
A)  $\frac { 6 } { 7 }$ 
B)  $\frac { 62 } { 21 }$ 
C)  $\frac { 64 } { 21 }$ 
D)  3

Question 4

Use integration by parts to establish a reduction formula for the integral.
-$$\int x ^ { n } e ^ { - x ^ { 2 } } d x$$

Accepted Answer

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

Question 5

Evaluate the integral.
-$$\int \frac { 3 e ^ { 4 t } + 12 e ^ { 2 t } + 4 } { e ^ { 2 t } + 4 } d t$$

Accepted Answer

A)  $\frac { 3 } { 2 } \mathrm { e } ^ { 2 \mathrm { t } } + \mathrm { t } - \frac { 1 } { 2 } \ln \left| \mathrm { e } ^ { 2 \mathrm { t } } + 4 \right| + \mathrm { C }$ 
B)  $\frac { 3 } { 2 } \mathrm { e } ^ { 2 \mathrm { t } } - \mathrm { t } + \ln \left| \mathrm { e } ^ { 2 \mathrm { t } } + 4 \right| + \mathrm { C }$ 
C)  $\mathrm { e } ^ { 2 \mathrm { t } } + \ln \left| \mathrm { e } ^ { \mathrm { t } } + 4 \right| - \frac { 1 } { 2 } \ln \left| \mathrm { e } ^ { 2 \mathrm { t } } \right| + \mathrm { C }$ 
D)  $\mathrm { e } ^ { 2 \mathrm { t } } + \mathrm { t } - \frac { 1 } { 2 } \ln \left| \mathrm { e } ^ { \mathrm { t } } + 4 \right| + \mathrm { C }$ 
A)  $\frac { 3 } { 2 } \mathrm { e } ^ { 2 \mathrm { t } } + \mathrm { t } - \frac { 1 } { 2 } \ln \left| \mathrm { e } ^ { 2 \mathrm { t } } + 4 \right| + \mathrm { C }$ 
B)  $\frac { 3 } { 2 } \mathrm { e } ^ { 2 \mathrm { t } } - \mathrm { t } + \ln \left| \mathrm { e } ^ { 2 \mathrm { t } } + 4 \right| + \mathrm { C }$ 
C)  $\mathrm { e } ^ { 2 \mathrm { t } } + \ln \left| \mathrm { e } ^ { \mathrm { t } } + 4 \right| - \frac { 1 } { 2 } \ln \left| \mathrm { e } ^ { 2 \mathrm { t } } \right| + \mathrm { C }$ 
D)  $\mathrm { e } ^ { 2 \mathrm { t } } + \mathrm { t } - \frac { 1 } { 2 } \ln \left| \mathrm { e } ^ { \mathrm { t } } + 4 \right| + \mathrm { C }$

Question 6

Solve the problem.
-A physical fitness association is including the mile run in its secondary-school fitness test. The time for this event for boys in secondary school is known to possess a normal distribution with a mean of 440 seconds and a
Standard deviation of 60 seconds. Find the probability that a randomly selected boy in secondary school can run
The mile in less than 302 seconds.

Accepted Answer

A)  0.4893 
B)  0.5107 
C)  0.0107 
D)  0.9893 
A)  0.4893 
B)  0.5107 
C)  0.0107 
D)  0.9893

Question 7

Evaluate the improper integral or state that it is divergent.
-$$\int _ { 1 } ^ { \infty } \frac { d x } { x ^ { 3.933 } }$$

Accepted Answer

A)  Divergent 
B)  $\frac { 1 } { 2.933 }$ 
C)  $\frac { 1 } { 3.933 }$ 
D)  $\frac { 1 } { 4.933 }$ 
A)  Divergent 
B)  $\frac { 1 } { 2.933 }$ 
C)  $\frac { 1 } { 3.933 }$ 
D)  $\frac { 1 } { 4.933 }$

Question 8

Express the integrand as a sum of partial fractions and evaluate the integral.
-$$\int \frac { 5 x ^ { 2 } + x + 36 } { x ^ { 3 } + 9 x } d x$$

Accepted Answer

A)  $4 \ln | x | + \frac { 1 } { 2 } \ln \left| x ^ { 2 } + 9 \right| + \frac { 1 } { 3 } \tan ^ { - 1 } \left( \frac { x } { 3 } \right) + C$ 
B)  $4 \ln | x | - \frac { 1 } { 2 } \ln \left| x ^ { 2 } + 9 \right| - \tan ^ { - 1 } x + C$ 
C)  $4 \ln | x | + \frac { 1 } { 2 } \ln \left| x ^ { 2 } + 9 \right| + \sin ^ { - 1 } \left( \frac { x } { 3 } \right) + C$ 
D)  $\ln | x | + \frac { 1 } { 2 } \ln \left| x ^ { 2 } + 9 \right| + \tan ^ { - 1 } \left( \frac { x } { 3 } \right) + C$ 
A)  $4 \ln | x | + \frac { 1 } { 2 } \ln \left| x ^ { 2 } + 9 \right| + \frac { 1 } { 3 } \tan ^ { - 1 } \left( \frac { x } { 3 } \right) + C$ 
B)  $4 \ln | x | - \frac { 1 } { 2 } \ln \left| x ^ { 2 } + 9 \right| - \tan ^ { - 1 } x + C$ 
C)  $4 \ln | x | + \frac { 1 } { 2 } \ln \left| x ^ { 2 } + 9 \right| + \sin ^ { - 1 } \left( \frac { x } { 3 } \right) + C$ 
D)  $\ln | x | + \frac { 1 } { 2 } \ln \left| x ^ { 2 } + 9 \right| + \tan ^ { - 1 } \left( \frac { x } { 3 } \right) + C$

Question 9

Evaluate the integral.
-$$\int \frac { d x } { x \sqrt { 6 + 4 x } }$$

Accepted Answer

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

Question 10

Find the area or volume.
-$\text { Find the area between the graph of } y = \frac { 12 } { ( x - 1 ) ^ { 2 } } \text { and the } x \text {-axis, for } - \infty < x \leq 0 \text {. }$

Accepted Answer

A)  6 
B)  1 
C)  12 
D)  24 
A)  6 
B)  1 
C)  12 
D)  24

Question 11

Solve the problem.
-The time between major earthquakes in the Alaska panhandle region can be modeled with an exponential distribution having a mean of 620 days. Find the probability that the time between a major earthquake and the
Next one is less than 250 days.

Accepted Answer

A)  0.6682 
B)  0.0011 
C)  0.0005 
D)  0.3318 
A)  0.6682 
B)  0.0011 
C)  0.0005 
D)  0.3318

Question 12

Evaluate the improper integral or state that it is divergent.
-$$\int _ { - \infty } ^ { \infty } x ^ { 3 } e ^ { - x ^ { 4 } } d x$$

Accepted Answer

A)  $- \frac { 1 } { 2 }$ 
B)  0 
C)  $\frac { 1 } { 4 }$ 
D)  Divergent 
A)  $- \frac { 1 } { 2 }$ 
B)  0 
C)  $\frac { 1 } { 4 }$ 
D)  Divergent

Question 13

Solve the problem.
-Suppose a brewery has a filling machine that fills 12 ounce bottles of beer. It is known that the amount of beer poured by this filling machine follows a normal distribution with a mean of 12.49 ounces and a standard
Deviation of 0.04 ounce. Find the probability that the bottle contains between 12.39 and 12.45 ounces.

Accepted Answer

A)  0.1649 
B)  0.1525 
C)  0.8351 
D)  0.8475 
A)  0.1649 
B)  0.1525 
C)  0.8351 
D)  0.8475

Question 14

Evaluate the integral.
-$\int \cot ^ { 4 } 3 x d x$

Accepted Answer

A)  $- \frac { 1 } { 9 } \cot ^ { 3 } 3 x + \frac { 1 } { 3 } \cot 3 x + x + C$ 
B)  $- \frac { 1 } { 3 } \cot ^ { 3 } 3 x + \cot 3 x + x + C$ 
C)  $- \frac { 1 } { 9 } \cot ^ { 3 } 3 x + \frac { 1 } { 3 } \cot 3 x + C$ 
D)  $- \frac { 1 } { 9 } \cot ^ { 3 } 3 x + \frac { 1 } { 3 } \cot ^ { 2 } 3 x + x + C$ 
A)  $- \frac { 1 } { 9 } \cot ^ { 3 } 3 x + \frac { 1 } { 3 } \cot 3 x + x + C$ 
B)  $- \frac { 1 } { 3 } \cot ^ { 3 } 3 x + \cot 3 x + x + C$ 
C)  $- \frac { 1 } { 9 } \cot ^ { 3 } 3 x + \frac { 1 } { 3 } \cot 3 x + C$ 
D)  $- \frac { 1 } { 9 } \cot ^ { 3 } 3 x + \frac { 1 } { 3 } \cot ^ { 2 } 3 x + x + C$

Question 15

Solve the problem by integration.
-Find the volume generated by rotating the area bounded by $y = \frac { 1 } { x ^ { 3 } + 6 x ^ { 2 } + 5 x } , x = 4 , x = 5$, and $y = 0$ about the $\mathrm { y }$-axis.

Accepted Answer

A)  $2 \pi \ln \frac { 25 } { 27 }$ 
B)  $\frac { 1 } { 2 } \pi \ln \frac { 3 } { 5 }$ 
C)  $\frac { 1 } { 2 } \pi \ln \frac { 27 } { 25 }$ 
D)  $\frac { 1 } { 4 } \pi \ln \frac { 27 } { 25 }$ 
A)  $2 \pi \ln \frac { 25 } { 27 }$ 
B)  $\frac { 1 } { 2 } \pi \ln \frac { 3 } { 5 }$ 
C)  $\frac { 1 } { 2 } \pi \ln \frac { 27 } { 25 }$ 
D)  $\frac { 1 } { 4 } \pi \ln \frac { 27 } { 25 }$

Question 16

Solve the problem.
-Estimate the minimum number of subintervals needed to approximate the integral $\int _ { 1 } ^ { 3 } \left( 6 x ^ { 4 } - 5 x \right) d x$ with an error of magnitude less than $10 ^ { - 4 }$ using Simpson's Rule.

Accepted Answer

A)  38 
B)  30 
C)  24 
D)  84 
A)  38 
B)  30 
C)  24 
D)  84

Question 17

Solve the problem.
-Suppose that the accompanying table shows the velocity of a car every second for 8 seconds. Use Simpson's Rule to approximate the distance traveled by the car in the 8 seconds. Round your answer to the nearest hundredth if necessary.
$\begin{array}{l|l}
\text { Time (sec) } & \text { Velocity (ft/sec) } \
\hline 0 & 18 \
1 & 19 \
2 & 20 \
3 & 22 \
4 & 21 \
5 & 23 \
6 & 20 \
7 & 18 \
8 & 19
\end{array}$

Accepted Answer

A)  160.33 ft 
B)  120 ft 
C)  162.33 ft 
D)  161.5 ft 
A)  160.33 ft 
B)  120 ft 
C)  162.33 ft 
D)  161.5 ft

Question 18

Use a trigonometric substitution to evaluate the integral.
-$$\int _ { 0 } ^ { 5 } \frac { 4 e ^ { - t } } { 1 + 16 e ^ { - 2 t } } d t$$

Accepted Answer

A)  $\tan ^ { - 1 } \frac { 4 } { e ^ { 5 } } - \tan ^ { - 1 } 4$ 
B)  $\tan ^ { - 1 } 4 - \tan ^ { - 1 } \frac { 4 } { e ^ { 5 } }$ 
C)  $\frac { 1 } { 4 } \tan ^ { - 1 } 4 - \frac { 1 } { 4 } \tan ^ { - 1 } \frac { 4 } { e ^ { 5 } }$ 
D)  $\tan ^ { - 1 } 5 - \tan ^ { - 1 } 4$ 
A)  $\tan ^ { - 1 } \frac { 4 } { e ^ { 5 } } - \tan ^ { - 1 } 4$ 
B)  $\tan ^ { - 1 } 4 - \tan ^ { - 1 } \frac { 4 } { e ^ { 5 } }$ 
C)  $\frac { 1 } { 4 } \tan ^ { - 1 } 4 - \frac { 1 } { 4 } \tan ^ { - 1 } \frac { 4 } { e ^ { 5 } }$ 
D)  $\tan ^ { - 1 } 5 - \tan ^ { - 1 } 4$

Question 19

Solve the problem.
-A physical fitness association is including the mile run in its secondary-school fitness test. The time for this event for boys in secondary school is known to possess a normal distribution with a mean of 450 seconds and a
Standard deviation of 60 seconds. Find the probability that a randomly selected boy in secondary school will
Take longer than 312 seconds to run the mile.

Accepted Answer

A)  0.0107 
B)  0.9893 
C)  0.5107 
D)  0.4893 
A)  0.0107 
B)  0.9893 
C)  0.5107 
D)  0.4893

Question 20

Evaluate the integral.
-$$\int \frac { d x } { \sqrt { x ^ { 2 } - 6 x - 7 } }$$

Accepted Answer

A)  $\frac { 1 } { 4 } \tan ^ { - 1 } \left( \frac { x - 3 } { 4 } \right) + C$ 
B)  $\sin ^ { - 1 } \left( \frac { x - 3 } { 4 } \right) + C$ 
C)  $\sin ^ { - 1 } \left( \frac { x + 3 } { 4 } \right) + C$ 
D)  $\sin ^ { - 1 } \left( \frac { x - 6 } { 4 } \right) + C$ 
A)  $\frac { 1 } { 4 } \tan ^ { - 1 } \left( \frac { x - 3 } { 4 } \right) + C$ 
B)  $\sin ^ { - 1 } \left( \frac { x - 3 } { 4 } \right) + C$ 
C)  $\sin ^ { - 1 } \left( \frac { x + 3 } { 4 } \right) + C$ 
D)  $\sin ^ { - 1 } \left( \frac { x - 6 } { 4 } \right) + C$

Expand the quotient by partial fractions. - $\frac { y + 2 } { y ^ { 2 } ( y + 1 ) }$

Integrate the function. - $\int _ { 0 } ^ { 1 } \frac { d x } { \sqrt { 81 - x ^ { 2 } } }$

For the given probability density function, over the stated interval, find the requested value. - $f ( x ) = \frac { 1 } { 7 } x$ , over $[ 1,4 ] ;$ Find the mean.

Use integration by parts to establish a reduction formula for the integral. - $\int x ^ { n } e ^ { - x ^ { 2 } } d x$

Evaluate the integral. - $\int \frac { 3 e ^ { 4 t } + 12 e ^ { 2 t } + 4 } { e ^ { 2 t } + 4 } d t$

Evaluate the improper integral or state that it is divergent. - $\int _ { 1 } ^ { \infty } \frac { d x } { x ^ { 3.933 } }$

Express the integrand as a sum of partial fractions and evaluate the integral. - $\int \frac { 5 x ^ { 2 } + x + 36 } { x ^ { 3 } + 9 x } d x$

Evaluate the integral. - $\int \frac { d x } { x \sqrt { 6 + 4 x } }$

Find the area or volume. - $\text { Find the area between the graph of } y = \frac { 12 } { ( x - 1 ) ^ { 2 } } \text { and the } x \text {-axis, for } - \infty < x \leq 0 \text {. }$

Solve the problem. -The time between major earthquakes in the Alaska panhandle region can be modeled with an exponential distribution having a mean of 620 days. Find the probability that the time between a major earthquake and the Next one is less than 250 days.

Evaluate the improper integral or state that it is divergent. - $\int _ { - \infty } ^ { \infty } x ^ { 3 } e ^ { - x ^ { 4 } } d x$

Evaluate the integral. - $\int \cot ^ { 4 } 3 x d x$

Solve the problem by integration. -Find the volume generated by rotating the area bounded by $y = \frac { 1 } { x ^ { 3 } + 6 x ^ { 2 } + 5 x } , x = 4 , x = 5$ , and $y = 0$ about the $\mathrm { y }$ -axis.

Solve the problem. -Estimate the minimum number of subintervals needed to approximate the integral $\int _ { 1 } ^ { 3 } \left( 6 x ^ { 4 } - 5 x \right) d x$ with an error of magnitude less than $10 ^ { - 4 }$ using Simpson's Rule.

Use a trigonometric substitution to evaluate the integral. - $\int _ { 0 } ^ { 5 } \frac { 4 e ^ { - t } } { 1 + 16 e ^ { - 2 t } } d t$

Evaluate the integral. - $\int \frac { d x } { \sqrt { x ^ { 2 } - 6 x - 7 } }$

Functions

Limits and Continuity

Derivatives

Applications of Derivatives

Integrals

Applications of Definite Integrals

Integrals and Transcendental Functions

First-Order Differential Equations

Infinite Sequences and Series

Parametric Equations and Polar Coordinates

Vectors and the Geometry of Space

Vector-Valued Functions and Motion in Space

Partial Derivatives

Multiple Integrals

Integrals and Vector Fields

Filters

Exam 9: Techniques of Integration

Expand the quotient by partial fractions. - y+2y2(y+1)\frac { y + 2 } { y ^ { 2 } ( y + 1 ) }y2(y+1)y+2​

Integrate the function. - ∫01dx81−x2\int _ { 0 } ^ { 1 } \frac { d x } { \sqrt { 81 - x ^ { 2 } } }∫01​81−x2​dx​

For the given probability density function, over the stated interval, find the requested value. - f(x)=17xf ( x ) = \frac { 1 } { 7 } xf(x)=71​x , over [1,4];[ 1,4 ] ;[1,4]; Find the mean.

Use integration by parts to establish a reduction formula for the integral. - ∫xne−x2dx\int x ^ { n } e ^ { - x ^ { 2 } } d x∫xne−x2dx

Evaluate the integral. - ∫3e4t+12e2t+4e2t+4dt\int \frac { 3 e ^ { 4 t } + 12 e ^ { 2 t } + 4 } { e ^ { 2 t } + 4 } d t∫e2t+43e4t+12e2t+4​dt

Evaluate the improper integral or state that it is divergent. - ∫1∞dxx3.933\int _ { 1 } ^ { \infty } \frac { d x } { x ^ { 3.933 } }∫1∞​x3.933dx​

Express the integrand as a sum of partial fractions and evaluate the integral. - ∫5x2+x+36x3+9xdx\int \frac { 5 x ^ { 2 } + x + 36 } { x ^ { 3 } + 9 x } d x∫x3+9x5x2+x+36​dx

Evaluate the integral. - ∫dxx6+4x\int \frac { d x } { x \sqrt { 6 + 4 x } }∫x6+4x​dx​