Question 1

Find the integral.
-$\int x^{2}\left(3 x+x^{-3}\right) d x$

Accepted Answer

A)  $\frac{3}{2} x^{4}-\ln |x|+C$ 
B)  $\frac{3}{4} x^{4}-\ln |x|+C$ 
C)  $\frac{3}{4} x^{4}+\ln |x|+C$ 
D)  $\frac{3}{2} x^{4}+\ln |x|+C$ 
A)  $\frac{3}{2} x^{4}-\ln |x|+C$ 
B)  $\frac{3}{4} x^{4}-\ln |x|+C$ 
C)  $\frac{3}{4} x^{4}+\ln |x|+C$ 
D)  $\frac{3}{2} x^{4}+\ln |x|+C$

Question 2

Approximate the area under the curve and above the $x$-axis using $n$ rectangles. Let the height of each rectangle be given by the value of the function at the right side of the rectangle.
-$f(x)=2 x^{2}+x+3$ from $x=-2$ to $x=1 ; n=3$

Accepted Answer

A)  21 
B)  17 
C)  13 
D)  25 
A)  21 
B)  17 
C)  13 
D)  25

Question 3

Provide an appropriate response.
-If we decide to use $u=x+e$ to find $(x+e)^{n} d x$, then which would be the correct substitution?
i. $\int u^{n}+1 d u$
ii. $\int u^{n} d x$
iii. $\int u^{n} d u$

Accepted Answer

A)  ii is correct. 
B)  $i$ is correct. 
C)  iii is correct. 
D)  None of these is correct. 
A)  ii is correct. 
B)  $i$ is correct. 
C)  iii is correct. 
D)  None of these is correct.

Question 4

Find the particular solution of the differential equation.
-$\frac{d y}{d x}=6 x^{2}-4 x+8 ; y=21$ when $x=1$

Accepted Answer

A)  $y=2 x^{3}-2 x^{2}+8 x+13$ 
B)  $y=2 x^{3}-2 x^{2}+8 x-13$ 
C)  $y=6 x^{3}-4 x^{2}+8 x+11$ 
D)  $y=2 x^{3}-4 x^{2}+8 x+15$ 
A)  $y=2 x^{3}-2 x^{2}+8 x+13$ 
B)  $y=2 x^{3}-2 x^{2}+8 x-13$ 
C)  $y=6 x^{3}-4 x^{2}+8 x+11$ 
D)  $y=2 x^{3}-4 x^{2}+8 x+15$

Question 5

Find the area of the shaded region.
-

Accepted Answer

A)  $\frac{5}{3}$ 
B)  $\frac{2}{3}$ 
C)  $\frac{4}{3}$ 
D)  $\frac{1}{3}$ 
A)  $\frac{5}{3}$ 
B)  $\frac{2}{3}$ 
C)  $\frac{4}{3}$ 
D)  $\frac{1}{3}$

Question 6

Solve the problem.
-Assume that the rate of change of the population of a certain city is given by $\frac{d y}{d t}=6000 e^{.06 t}$, where $y$ is the population at time $t$, in years. The population was 100,000 in 1980 . Assume $t=0$ is 1980. Predict the population in 1990 .

Accepted Answer

A)  $1.8 \times 10^{5}$ 
B)  $1.9 \times 10^{5}$ 
C)  $9.2 \times 10^{5}$ 
D)  $1.1 \times 10^{5}$ 
A)  $1.8 \times 10^{5}$ 
B)  $1.9 \times 10^{5}$ 
C)  $9.2 \times 10^{5}$ 
D)  $1.1 \times 10^{5}$

Question 7

Evaluate the integral.
-$\int_{1}^{4}\left(x^{3 / 2}+x^{1 / 2}-x^{-1 / 2}\right) d x$

Accepted Answer

A)  46 
B)  14.67 
C)  14.93 
D)  15.07 
A)  46 
B)  14.67 
C)  14.93 
D)  15.07

Question 8

Find the integral.
-$\int \frac{\mathrm{e}^{\mathrm{x}} \mathrm{dx}}{\mathrm{e}^{\mathrm{x}}+\mathrm{e}}$

Accepted Answer

A)  $x+C$ 
B)  $\frac{x}{e}+C$ 
C)  $e \ln \left|e^{x}+e\right|+C$ 
D)  $\ln \left|e^{x}+e\right|+C$ 
A)  $x+C$ 
B)  $\frac{x}{e}+C$ 
C)  $e \ln \left|e^{x}+e\right|+C$ 
D)  $\ln \left|e^{x}+e\right|+C$

Question 9

Solve the problem.
-The rate of change in the number of bacteria (in thousands) in a culture after the introduction of a bactericide is $\mathrm{dy} / \mathrm{dt}=100-\mathrm{y}$, where $\mathrm{y}$ is the number of bacteria (in thousands) at time $t$. How many bacteria are present at $t=12$ if the initial population was $1,000,000$ thousand? (Round results to the nearest thousand bacteria.)

Accepted Answer

A)  $162,739 \times 10^{6}$ thousand bacteria 
B)  94 thousand bacteria 
C)  106 thousand bacteria 
D)  6 thousand bacteria 
A)  $162,739 \times 10^{6}$ thousand bacteria 
B)  94 thousand bacteria 
C)  106 thousand bacteria 
D)  6 thousand bacteria

Question 10

Find the integral.
-$\int 11 x^{2} e^{\left(-4 x^{3}\right)} d x$

Accepted Answer

A)  $\frac{11}{12} e^{\left(-4 x^{3}\right)}+C$ 
B)  $-11 e^{\left(-4 x^{3}\right)}+C$ 
C)  $12 e^{\left(-4 x^{3}\right)}+C$ 
D)  $-\frac{11}{12} e^{\left(-4 x^{3}\right)}+C$ 
A)  $\frac{11}{12} e^{\left(-4 x^{3}\right)}+C$ 
B)  $-11 e^{\left(-4 x^{3}\right)}+C$ 
C)  $12 e^{\left(-4 x^{3}\right)}+C$ 
D)  $-\frac{11}{12} e^{\left(-4 x^{3}\right)}+C$

Question 11

Use the numeric integration feature on your calculator to find the integral.
-$\int_{1}^{6} t \ln (t) d t$

Accepted Answer

A)  23.50167 
B)  None of the above is correct. 
C)  0.50167 
D)  2.350167 
A)  23.50167 
B)  None of the above is correct. 
C)  0.50167 
D)  2.350167

Question 12

Find the area bounded by the given curves.
-$y=x^{3}, y=4 x$

Accepted Answer

A)  4 
B)  8 
C)  16 
D)  2 
A)  4 
B)  8 
C)  16 
D)  2

Question 13

Evaluate.
-$\int 4 x^{2 / 3} d x$

Accepted Answer

A)  $\frac{8}{3} x^{5 / 3}+$ C 
B)  $\frac{4}{3} x^{5 / 3}+C$ 
C)  $\frac{12}{5} x^{5 / 3}+C$ 
D)  $6 x^{5 / 3}+C$ 
A)  $\frac{8}{3} x^{5 / 3}+$ C 
B)  $\frac{4}{3} x^{5 / 3}+C$ 
C)  $\frac{12}{5} x^{5 / 3}+C$ 
D)  $6 x^{5 / 3}+C$

Question 14

Find the area bounded by the given curves.
-$y=x, y=x^{2}$

Accepted Answer

A)  $\frac{1}{12}$ 
B)  $\frac{1}{2}$ 
C)  $\frac{1}{3}$ 
D)  $\frac{1}{6}$ 
A)  $\frac{1}{12}$ 
B)  $\frac{1}{2}$ 
C)  $\frac{1}{3}$ 
D)  $\frac{1}{6}$

Question 15

Find the general solution of the differential equation.
-$\frac{d y}{d x}=7 e^{3 x}$

Accepted Answer

A)  $7 e^{3 x}+C$ 
B)  $21 e^{3 x}+C$ 
C)  $\frac{7}{3} e^{3 x}+C$ 
D)  $\frac{1}{3} e^{3 x}+C$ 
A)  $7 e^{3 x}+C$ 
B)  $21 e^{3 x}+C$ 
C)  $\frac{7}{3} e^{3 x}+C$ 
D)  $\frac{1}{3} e^{3 x}+C$

Question 16

Use integration by parts to find the integral.
-$\int(x-6) e^{4 x} d x$

Accepted Answer

A)  $4(x-6) e^{4 x}-16 e^{4 x}+C$ 
B)  $\frac{1}{4}(x-6) e^{4 x}+\frac{1}{16} e^{4 x}+C$ 
C)  $\frac{1}{4}(x-6) e^{4 x}-\frac{1}{16} e^{4 x}+C$ 
D)  $(x-6) e^{4 x}-e^{4 x}+C$ 
A)  $4(x-6) e^{4 x}-16 e^{4 x}+C$ 
B)  $\frac{1}{4}(x-6) e^{4 x}+\frac{1}{16} e^{4 x}+C$ 
C)  $\frac{1}{4}(x-6) e^{4 x}-\frac{1}{16} e^{4 x}+C$ 
D)  $(x-6) e^{4 x}-e^{4 x}+C$

Question 17

Evaluate the integral.
-$\int_{-1}^{3}(x+5) d x$

Accepted Answer

A)  -24 
B)  15 
C)  24 
D)  10 
A)  -24 
B)  15 
C)  24 
D)  10

Question 18

Use the numeric integration feature on your calculator to find the integral.
-$\int_{2}^{4} \mathrm{e}^{-\mathrm{t}^{2}} \mathrm{dt}$

Accepted Answer

A)  0.05048 
B)  0.00248 
C)  0.00415 
D)  0.88623 
A)  0.05048 
B)  0.00248 
C)  0.00415 
D)  0.88623

Question 19

Solve the problem.
-The rate of water usage for a business, in gallons per day, is given by $W(t)=663 \mathrm{te}^{-t}$, where $t$ represents the number of hours since midnight. Approximately how many gallons of water does the business use in the first 6 hours of the day?

Accepted Answer

A)  655 gallons 
B)  675 gallons 
C)  12 gallons 
D)  651 gallons 
A)  655 gallons 
B)  675 gallons 
C)  12 gallons 
D)  651 gallons

Question 20

Evaluate the integral.
-$\int_{1}^{3}\left(2 x^{3}-5 x^{-2}\right) d x$

Accepted Answer

A)  36.67 
B)  50 
C)  60 
D)  47.17 
A)  36.67 
B)  50 
C)  60 
D)  47.17

Find the integral. - $\int x^{2}\left(3 x+x^{-3}\right) d x$

Approximate the area under the curve and above the $x$ -axis using $n$ rectangles. Let the height of each rectangle be given by the value of the function at the right side of the rectangle. - $f(x)=2 x^{2}+x+3$ from $x=-2$ to $x=1 ; n=3$

Provide an appropriate response. -If we decide to use $u=x+e$ to find $(x+e)^{n} d x$ , then which would be the correct substitution? i. $\int u^{n}+1 d u$ ii. $\int u^{n} d x$ iii. $\int u^{n} d u$

Find the particular solution of the differential equation. - $\frac{d y}{d x}=6 x^{2}-4 x+8 ; y=21$ when $x=1$

Find the area of the shaded region. -

Solve the problem. -Assume that the rate of change of the population of a certain city is given by $\frac{d y}{d t}=6000 e^{.06 t}$ , where $y$ is the population at time $t$ , in years. The population was 100,000 in 1980 . Assume $t=0$ is 1980. Predict the population in 1990 .

Evaluate the integral. - $\int_{1}^{4}\left(x^{3 / 2}+x^{1 / 2}-x^{-1 / 2}\right) d x$

Find the integral. - $\int \frac{\mathrm{e}^{\mathrm{x}} \mathrm{dx}}{\mathrm{e}^{\mathrm{x}}+\mathrm{e}}$

Find the integral. - $\int 11 x^{2} e^{\left(-4 x^{3}\right)} d x$

Use the numeric integration feature on your calculator to find the integral. - $\int_{1}^{6} t \ln (t) d t$

Find the area bounded by the given curves. - $y=x^{3}, y=4 x$

Evaluate. - $\int 4 x^{2 / 3} d x$

Find the area bounded by the given curves. - $y=x, y=x^{2}$

Find the general solution of the differential equation. - $\frac{d y}{d x}=7 e^{3 x}$

Use integration by parts to find the integral. - $\int(x-6) e^{4 x} d x$

Evaluate the integral. - $\int_{-1}^{3}(x+5) d x$

Use the numeric integration feature on your calculator to find the integral. - $\int_{2}^{4} \mathrm{e}^{-\mathrm{t}^{2}} \mathrm{dt}$

Solve the problem. -The rate of water usage for a business, in gallons per day, is given by $W(t)=663 \mathrm{te}^{-t}$ , where $t$ represents the number of hours since midnight. Approximately how many gallons of water does the business use in the first 6 hours of the day?

Evaluate the integral. - $\int_{1}^{3}\left(2 x^{3}-5 x^{-2}\right) d x$

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Exam 13: Integral Calculus

Find the integral. - ∫x2(3x+x−3)dx\int x^{2}\left(3 x+x^{-3}\right) d x∫x2(3x+x−3)dx

Approximate the area under the curve and above the xxx -axis using nnn rectangles. Let the height of each rectangle be given by the value of the function at the right side of the rectangle. - f(x)=2x2+x+3f(x)=2 x^{2}+x+3f(x)=2x2+x+3 from x=−2x=-2x=−2 to x=1;n=3x=1 ; n=3x=1;n=3

Provide an appropriate response. -If we decide to use u=x+eu=x+eu=x+e to find (x+e)ndx(x+e)^{n} d x(x+e)ndx , then which would be the correct substitution? i. ∫un+1du\int u^{n}+1 d u∫un+1du ii. ∫undx\int u^{n} d x∫undx iii. ∫undu\int u^{n} d u∫undu

Find the particular solution of the differential equation. - dydx=6x2−4x+8;y=21\frac{d y}{d x}=6 x^{2}-4 x+8 ; y=21dxdy​=6x2−4x+8;y=21 when x=1x=1x=1