Question 1

Evaluate.
-$\int \frac{5}{\mathrm{x}^{2}}-\frac{4}{\sqrt{\mathrm{x}}} \mathrm{dx}$

Accepted Answer

A)  $-\frac{5}{x}-8 \sqrt{x}+C$ 
B)  $-\frac{5}{x}-\frac{8}{\sqrt{x}}+C$ 
C)  $\frac{5}{x}-8 \sqrt{x}-C$ 
D)  $\frac{5}{x}-\frac{8}{\sqrt{x}}+C$ 
A)  $-\frac{5}{x}-8 \sqrt{x}+C$ 
B)  $-\frac{5}{x}-\frac{8}{\sqrt{x}}+C$ 
C)  $\frac{5}{x}-8 \sqrt{x}-C$ 
D)  $\frac{5}{x}-\frac{8}{\sqrt{x}}+C$

Question 2

Estimate the value of the quantity
-The graph below shows the rate of change of the price of a stock (in dollars per share per week) over a period of 6 weeks. Estimate the total change in dollars per share of the stock during this period. Use rectangles with widths of 1 week and let the function value at the midpoint of the rectangle give the height of the rectangle.

Accepted Answer

A)  $\$ 1.60 /$ share 
B)  $\$ 0.90 /$ share 
C)  $\$ 2.00 /$ share 
D)  $\$ 2.20 /$ share 
A)  $\$ 1.60 /$ share 
B)  $\$ 0.90 /$ share 
C)  $\$ 2.00 /$ share 
D)  $\$ 2.20 /$ share

Question 3

Find the integral.
-$\int 9 z \sqrt{3 z^{2}-7} d z$

Accepted Answer

A)  $\frac{1}{2} z\left(3 z^{2}-7\right)^{3 / 2}+C$ 
B)  $\left(3 z^{2}-7\right)^{3 / 2}+C$ 
C)  $z\left(3 z^{2}-7\right)^{3 / 2}+C$ 
D)  $\frac{1}{2}\left(3 z^{2}-7\right)^{3 / 2}+C$ 
A)  $\frac{1}{2} z\left(3 z^{2}-7\right)^{3 / 2}+C$ 
B)  $\left(3 z^{2}-7\right)^{3 / 2}+C$ 
C)  $z\left(3 z^{2}-7\right)^{3 / 2}+C$ 
D)  $\frac{1}{2}\left(3 z^{2}-7\right)^{3 / 2}+C$

Question 4

Find the given indefinite integral. State whether integration by substitution or integration by parts was used.
-$\int 11 \mathrm{xe}^{\mathrm{x}^{2}+1} \mathrm{dx}$

Accepted Answer

A)  $22 \mathrm{e}^{\mathrm{x}^{2}+1}-\frac{11}{2} \mathrm{e}^{\mathrm{x}^{2}+1}+\mathrm{C}$; parts 
B)  $22 \mathrm{e}^{\mathrm{x}^{2}+1}+\mathrm{C}$; substitution 
C)  $\frac{11}{2} \mathrm{e}^{\mathrm{x}^{2}+1}+\mathrm{C}$; substitution 
D)  $121 \mathrm{e}^{\mathrm{x}^{2}+1}-\frac{11}{2} \mathrm{e}^{\mathrm{x}^{2}+1}+\mathrm{C}$; parts 
A)  $22 \mathrm{e}^{\mathrm{x}^{2}+1}-\frac{11}{2} \mathrm{e}^{\mathrm{x}^{2}+1}+\mathrm{C}$; parts 
B)  $22 \mathrm{e}^{\mathrm{x}^{2}+1}+\mathrm{C}$; substitution 
C)  $\frac{11}{2} \mathrm{e}^{\mathrm{x}^{2}+1}+\mathrm{C}$; substitution 
D)  $121 \mathrm{e}^{\mathrm{x}^{2}+1}-\frac{11}{2} \mathrm{e}^{\mathrm{x}^{2}+1}+\mathrm{C}$; parts

Question 5

Find the integral.
-$\int \frac{19}{2+5 y} d y$

Accepted Answer

A)  $\frac{19}{5} \ln |2+5 y|+C$ 
B)  $18 \ln |2+5 y|+C$ 
C)  $\frac{18}{5} \ln |2+5 y|+C$ 
D)  $19 \ln |2+5 y|+C$ 
A)  $\frac{19}{5} \ln |2+5 y|+C$ 
B)  $18 \ln |2+5 y|+C$ 
C)  $\frac{18}{5} \ln |2+5 y|+C$ 
D)  $19 \ln |2+5 y|+C$

Question 6

Evaluate the integral.
-$\int_{-2}^{-1} 2 x^{-4} d x$

Accepted Answer

A)  $\frac{7}{24}$ 
B)  $\frac{1}{12}$ 
C)  $\frac{7}{12}$ 
D)  14 
A)  $\frac{7}{24}$ 
B)  $\frac{1}{12}$ 
C)  $\frac{7}{12}$ 
D)  14

Question 7

Use the definite integral to find the area between the $x$-axis and the graph of $f(x)$ over the indicated interval.
-$\mathrm{F}(\mathrm{x})=\mathrm{x}^{-(1 / 2)} ;[1,4]$

Accepted Answer

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

Question 8

A student has a supply equation and selling price. What kind of surplus can the student calculate with this information?

Accepted Answer

A)  The producers' surplus 
B)  The consumers' surplus 
C)  Both the consumers' and producers' surpluses 
A)  The producers' surplus 
B)  The consumers' surplus 
C)  Both the consumers' and producers' surpluses

Question 9

Evaluate.
-$\int 12 e^{-0.2 x} d x$

Accepted Answer

A)  $\frac{12 e^{-0.2 x+1}}{-0.2 x+1}+C$ 
B)  $-2.4 e^{-0.2 x}+C$ 
C)  $12 e^{-0.2 x}+C$ 
D)  $-60 e^{-0.2 x}+C$ 
A)  $\frac{12 e^{-0.2 x+1}}{-0.2 x+1}+C$ 
B)  $-2.4 e^{-0.2 x}+C$ 
C)  $12 e^{-0.2 x}+C$ 
D)  $-60 e^{-0.2 x}+C$

Question 10

Evaluate.
-$\int 7 x^{-3} d x$

Accepted Answer

A)  $7 x^{3}$ 
B)  $-21 x+C$ 
C)  $x+7 C$ 
D)  $-\frac{7}{2 x^{2}}+C$ 
A)  $7 x^{3}$ 
B)  $-21 x+C$ 
C)  $x+7 C$ 
D)  $-\frac{7}{2 x^{2}}+C$

Question 11

Solve the problem.
-Suppose that an object's acceleration function is given by $a(t)=10 t+6$. The object's initial velocity, $v(0)$, is 3 , and the object's initial position, $s(0)$, is 12 . Find $s(t)$.

Accepted Answer

A)  $\frac{10}{3} t^{3}+3 t^{2}+12 t+3$ 
B)  $\frac{5}{3} t^{3}+3 t^{2}+3 t+12$ 
C)  $5 t^{2}+6 t+3$ 
D)  $\frac{5}{3} t^{3}+3 t^{2}+3 t$ 
A)  $\frac{10}{3} t^{3}+3 t^{2}+12 t+3$ 
B)  $\frac{5}{3} t^{3}+3 t^{2}+3 t+12$ 
C)  $5 t^{2}+6 t+3$ 
D)  $\frac{5}{3} t^{3}+3 t^{2}+3 t$

Question 12

Evaluate the definite integral.
-$\int_{-8}^{8} e^{x / 8} d x$

Accepted Answer

A)  $e+e^{-1}$ 
B)  $\frac{1}{2}\left(e-e^{-1}\right)$ 
C)  $e-e^{-1}$ 
D)  $8\left(e-e^{-1}\right)$ 
A)  $e+e^{-1}$ 
B)  $\frac{1}{2}\left(e-e^{-1}\right)$ 
C)  $e-e^{-1}$ 
D)  $8\left(e-e^{-1}\right)$

Question 13

Solve the problem.
-Suppose the supply function of a certain item is given by $S(x)=2 x+7$ and the demand function is $D(x)=27-x^{2 / 3}$. Find the consumer's surplus.

Accepted Answer

A)  $\frac{32}{5}$ 
B)  32 
C)  $\frac{64}{5}$ 
D)  64 
A)  $\frac{32}{5}$ 
B)  32 
C)  $\frac{64}{5}$ 
D)  64

Question 14

Solve the problem.
-For a certain drug, the rate of reaction in appropriate units is given by $R^{\prime}(t)=\frac{4}{t}+\frac{5}{t^{2}}$, where $t$ is measured in hours after the drug is administered. Find the total reaction to the drug from $t=2$ to $t=7$.

Accepted Answer

A)  3.67 
B)  6.8 
C)  7.34 
D)  10.6 
A)  3.67 
B)  6.8 
C)  7.34 
D)  10.6

Question 15

Find the particular solution of the differential equation.
-$\frac{d y}{d x}=4 x e^{2 x} ; y=22$ when $x=0$

Accepted Answer

A)  $y=4 x e^{2 x}-e^{2 x}+23$ 
B)  $y=2 x e^{2 x}-e^{2 x}+23$ 
C)  $y=4 x e^{2 x}-2 e^{2 x}+24$ 
D)  $y=2 x e^{2 x}+22$ 
A)  $y=4 x e^{2 x}-e^{2 x}+23$ 
B)  $y=2 x e^{2 x}-e^{2 x}+23$ 
C)  $y=4 x e^{2 x}-2 e^{2 x}+24$ 
D)  $y=2 x e^{2 x}+22$

Question 16

Solve the problem.
-Find the consumer's surplus if the demand for an item is given by $D(x)=72-x^{2}$, assuming supply and demand are in equilibrium at $x=6$.

Accepted Answer

A)  216 
B)  432 
C)  72 
D)  144 
A)  216 
B)  432 
C)  72 
D)  144

Question 17

Solve the problem.
-The velocity $\mathrm{v}$ (in $\mathrm{m} / \mathrm{s}$ ) of an object moving with acceleration a (in $\mathrm{m} / \mathrm{s}^{2}$ ) is given by $\mathrm{v}=\int \mathrm{adt}$, where $t$ is the time (in seconds). Find a formula for $v$, if $a=\frac{5}{4} t\left(t^{2}+1\right)^{2}$.

Accepted Answer

A)  $v=\frac{1}{2} \frac{\left(t^{2}+1\right)^{3}}{3}+C$ 
B)  $v=\frac{5}{8} \frac{\left(t^{2}+1\right)^{3}}{3}+C$ 
C)  $\mathrm{v}=\frac{1}{4} \frac{\left(\mathrm{t}^{2}+1\right)^{3}}{3}+\mathrm{C}$ 
D)  $\mathrm{v}=-\frac{1}{8} \frac{\left(\mathrm{t}^{2}+1\right)^{3}}{3}+\mathrm{C}$ 
A)  $v=\frac{1}{2} \frac{\left(t^{2}+1\right)^{3}}{3}+C$ 
B)  $v=\frac{5}{8} \frac{\left(t^{2}+1\right)^{3}}{3}+C$ 
C)  $\mathrm{v}=\frac{1}{4} \frac{\left(\mathrm{t}^{2}+1\right)^{3}}{3}+\mathrm{C}$ 
D)  $\mathrm{v}=-\frac{1}{8} \frac{\left(\mathrm{t}^{2}+1\right)^{3}}{3}+\mathrm{C}$

Question 18

Find the integral.
-$\int \frac{\sqrt{\mathrm{x}}-6}{2 \mathrm{x} \sqrt{\mathrm{x}}} \mathrm{dx}$

Accepted Answer

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

Question 19

Solve the problem.
-The slope to the tangent line of a curve is given by $f^{\prime}(x)=x^{2}-11 x+7$. If the point $(0,5)$ is on the curve, find an equation of the curve.

Accepted Answer

A) f $(x)=\frac{1}{3} x^{3}-12 x^{2}+7 x+5$ 
B)  $f(x)=\frac{1}{3} x^{3}-\frac{11}{2} x^{2}+7 x+5$ 
C)  $f(x)=\frac{1}{3} x^{3}-\frac{11}{2} x^{2}+7 x+1$ 
D)  $f(x)=\frac{1}{3} x^{3}-12 x^{2}+7 x+1$ 
A) f $(x)=\frac{1}{3} x^{3}-12 x^{2}+7 x+5$ 
B)  $f(x)=\frac{1}{3} x^{3}-\frac{11}{2} x^{2}+7 x+5$ 
C)  $f(x)=\frac{1}{3} x^{3}-\frac{11}{2} x^{2}+7 x+1$ 
D)  $f(x)=\frac{1}{3} x^{3}-12 x^{2}+7 x+1$

Question 20

Evaluate the integral.
-$\int_{1}^{5} 6 x^{-2} d x$

Accepted Answer

A)  4.8 
B)  -7.2 
C)  11.9 
D)  -6 
A)  4.8 
B)  -7.2 
C)  11.9 
D)  -6

Evaluate. - $\int \frac{5}{\mathrm{x}^{2}}-\frac{4}{\sqrt{\mathrm{x}}} \mathrm{dx}$

Find the integral. - $\int 9 z \sqrt{3 z^{2}-7} d z$

Find the given indefinite integral. State whether integration by substitution or integration by parts was used. - $\int 11 \mathrm{xe}^{\mathrm{x}^{2}+1} \mathrm{dx}$

Find the integral. - $\int \frac{19}{2+5 y} d y$

Evaluate the integral. - $\int_{-2}^{-1} 2 x^{-4} d x$

Use the definite integral to find the area between the $x$ -axis and the graph of $f(x)$ over the indicated interval. - $\mathrm{F}(\mathrm{x})=\mathrm{x}^{-(1 / 2)} ;[1,4]$

A student has a supply equation and selling price. What kind of surplus can the student calculate with this information?

Evaluate. - $\int 12 e^{-0.2 x} d x$

Evaluate. - $\int 7 x^{-3} d x$

Solve the problem. -Suppose that an object's acceleration function is given by $a(t)=10 t+6$ . The object's initial velocity, $v(0)$ , is 3 , and the object's initial position, $s(0)$ , is 12 . Find $s(t)$ .

Evaluate the definite integral. - $\int_{-8}^{8} e^{x / 8} d x$

Solve the problem. -Suppose the supply function of a certain item is given by $S(x)=2 x+7$ and the demand function is $D(x)=27-x^{2 / 3}$ . Find the consumer's surplus.

Solve the problem. -For a certain drug, the rate of reaction in appropriate units is given by $R^{\prime}(t)=\frac{4}{t}+\frac{5}{t^{2}}$ , where $t$ is measured in hours after the drug is administered. Find the total reaction to the drug from $t=2$ to $t=7$ .

Find the particular solution of the differential equation. - $\frac{d y}{d x}=4 x e^{2 x} ; y=22$ when $x=0$

Solve the problem. -Find the consumer's surplus if the demand for an item is given by $D(x)=72-x^{2}$ , assuming supply and demand are in equilibrium at $x=6$ .

Find the integral. - $\int \frac{\sqrt{\mathrm{x}}-6}{2 \mathrm{x} \sqrt{\mathrm{x}}} \mathrm{dx}$

Solve the problem. -The slope to the tangent line of a curve is given by $f^{\prime}(x)=x^{2}-11 x+7$ . If the point $(0,5)$ is on the curve, find an equation of the curve.

Evaluate the integral. - $\int_{1}^{5} 6 x^{-2} d x$

Algebra and Equations

Graphs, Lines, and Inequalities

Functions and Graphs

Exponential and Logarithmic Functions

Mathematics of Finance

Systems of Linear Equations and Matrices

Linear Programming

Sets and Probability

Counting, Probability Distributions, and Further Topics in Probability

Introduction to Statistics

Differential Calculus

Applications of the Derivative

Multivariate Calculus

Filters

Exam 13: Integral Calculus

Evaluate. - ∫5x2−4xdx\int \frac{5}{\mathrm{x}^{2}}-\frac{4}{\sqrt{\mathrm{x}}} \mathrm{dx}∫x25​−x​4​dx

Find the integral. - ∫9z3z2−7dz\int 9 z \sqrt{3 z^{2}-7} d z∫9z3z2−7​dz

Find the given indefinite integral. State whether integration by substitution or integration by parts was used. - ∫11xex2+1dx\int 11 \mathrm{xe}^{\mathrm{x}^{2}+1} \mathrm{dx}∫11xex2+1dx

Find the integral. - ∫192+5ydy\int \frac{19}{2+5 y} d y∫2+5y19​dy

Evaluate the integral. - ∫−2−12x−4dx\int_{-2}^{-1} 2 x^{-4} d x∫−2−1​2x−4dx

Use the definite integral to find the area between the xxx -axis and the graph of f(x)f(x)f(x) over the indicated interval. - F(x)=x−(1/2);[1,4]\mathrm{F}(\mathrm{x})=\mathrm{x}^{-(1 / 2)} ;[1,4]F(x)=x−(1/2);[1,4]

A student has a supply equation and selling price. What kind of surplus can the student calculate with this information?

Evaluate. - ∫12e−0.2xdx\int 12 e^{-0.2 x} d x∫12e−0.2xdx

Evaluate. - ∫7x−3dx\int 7 x^{-3} d x∫7x−3dx

Solve the problem. -Suppose that an object's acceleration function is given by a(t)=10t+6a(t)=10 t+6a(t)=10t+6 . The object's initial velocity, v(0)v(0)v(0) , is 3 , and the object's initial position, s(0)s(0)s(0) , is 12 . Find s(t)s(t)s(t) .

Evaluate the definite integral. - ∫−88ex/8dx\int_{-8}^{8} e^{x / 8} d x∫−88​ex/8dx

Solve the problem. -Suppose the supply function of a certain item is given by S(x)=2x+7S(x)=2 x+7S(x)=2x+7 and the demand function is D(x)=27−x2/3D(x)=27-x^{2 / 3}D(x)=27−x2/3 . Find the consumer's surplus.

Find the particular solution of the differential equation. - dydx=4xe2x;y=22\frac{d y}{d x}=4 x e^{2 x} ; y=22dxdy​=4xe2x;y=22 when x=0x=0x=0

Solve the problem. -Find the consumer's surplus if the demand for an item is given by D(x)=72−x2D(x)=72-x^{2}D(x)=72−x2 , assuming supply and demand are in equilibrium at x=6x=6x=6 .

Find the integral. - ∫x−62xxdx\int \frac{\sqrt{\mathrm{x}}-6}{2 \mathrm{x} \sqrt{\mathrm{x}}} \mathrm{dx}∫2xx​x​−6​dx

Solve the problem. -The slope to the tangent line of a curve is given by f′(x)=x2−11x+7f^{\prime}(x)=x^{2}-11 x+7f′(x)=x2−11x+7 . If the point (0,5)(0,5)(0,5) is on the curve, find an equation of the curve.

Evaluate the integral. - ∫156x−2dx\int_{1}^{5} 6 x^{-2} d x∫15​6x−2dx

Algebra and Equations

Graphs, Lines, and Inequalities

Functions and Graphs

Exponential and Logarithmic Functions

Mathematics of Finance

Systems of Linear Equations and Matrices

Linear Programming

Sets and Probability

Counting, Probability Distributions, and Further Topics in Probability

Introduction to Statistics

Differential Calculus

Applications of the Derivative

Multivariate Calculus

Filters

Evaluate. - $\int \frac{5}{\mathrm{x}^{2}}-\frac{4}{\sqrt{\mathrm{x}}} \mathrm{dx}$

Find the integral. - $\int 9 z \sqrt{3 z^{2}-7} d z$

Find the given indefinite integral. State whether integration by substitution or integration by parts was used. - $\int 11 \mathrm{xe}^{\mathrm{x}^{2}+1} \mathrm{dx}$

Find the integral. - $\int \frac{19}{2+5 y} d y$

Evaluate the integral. - $\int_{-2}^{-1} 2 x^{-4} d x$

Use the definite integral to find the area between the $x$ -axis and the graph of $f(x)$ over the indicated interval. - $\mathrm{F}(\mathrm{x})=\mathrm{x}^{-(1 / 2)} ;[1,4]$

Evaluate. - $\int 12 e^{-0.2 x} d x$

Evaluate. - $\int 7 x^{-3} d x$

Solve the problem. -Suppose that an object's acceleration function is given by $a(t)=10 t+6$ . The object's initial velocity, $v(0)$ , is 3 , and the object's initial position, $s(0)$ , is 12 . Find $s(t)$ .

Evaluate the definite integral. - $\int_{-8}^{8} e^{x / 8} d x$

Solve the problem. -Suppose the supply function of a certain item is given by $S(x)=2 x+7$ and the demand function is $D(x)=27-x^{2 / 3}$ . Find the consumer's surplus.

Find the particular solution of the differential equation. - $\frac{d y}{d x}=4 x e^{2 x} ; y=22$ when $x=0$

Solve the problem. -Find the consumer's surplus if the demand for an item is given by $D(x)=72-x^{2}$ , assuming supply and demand are in equilibrium at $x=6$ .

Find the integral. - $\int \frac{\sqrt{\mathrm{x}}-6}{2 \mathrm{x} \sqrt{\mathrm{x}}} \mathrm{dx}$

Solve the problem. -The slope to the tangent line of a curve is given by $f^{\prime}(x)=x^{2}-11 x+7$ . If the point $(0,5)$ is on the curve, find an equation of the curve.

Evaluate the integral. - $\int_{1}^{5} 6 x^{-2} d x$