Question 1

Evaluate the definite integral.
-$\int_{0}^{3}(4 x+3)^{3} d x$

Accepted Answer

A)  3164.06 
B)  3159 
C)  12,636 
D)  50,544 
A)  3164.06 
B)  3159 
C)  12,636 
D)  50,544

Question 2

Use the definite integral to find the area between the $x$-axis and the graph of $f(x)$ over the indicated interval.
-$f(x)=\frac{3}{x^{3}} ;[1,3]$

Accepted Answer

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

Question 3

Find the particular solution of the differential equation.
-$\frac{d y}{d x}+y=2 e^{x} ; y=21$ when $x=0$

Accepted Answer

A)  $y=e^{x}+20 e^{-x}$ 
B)  $y=4 e^{2}+20 e^{-x}$ 
C)  $y=21 e^{x}$ 
D)  $y=2 e^{x}+18 e^{-x}$ 
A)  $y=e^{x}+20 e^{-x}$ 
B)  $y=4 e^{2}+20 e^{-x}$ 
C)  $y=21 e^{x}$ 
D)  $y=2 e^{x}+18 e^{-x}$

Question 4

Solve the problem.
-A certain company has found that its expenditure rate per day (in hundreds of dollars) on a certain type of job is given by $E^{\prime}(x)=8 x+7$, where $x$ is the number of days since the start of the job. Find the expenditure if the job takes 5 days.

Accepted Answer

A)  $\$ 47$ 
B)  $\$ 4700$ 
C)  $\$ 13,500$ 
D)  $\$ 135$ 
A)  $\$ 47$ 
B)  $\$ 4700$ 
C)  $\$ 13,500$ 
D)  $\$ 135$

Question 5

Solve the problem.
-For a certain drug, the rate of reaction in appropriate units is given by $R^{\prime}(t)=\frac{3}{t}+\frac{2}{t^{2}}$, where $t$ is measured in hours after the drug is administered. Find the total reaction to the drug from $t=2$ to $\mathrm{t}=8$.

Accepted Answer

A)  4.91 
B)  8.98 
C)  3.54 
D)  7.07 
A)  4.91 
B)  8.98 
C)  3.54 
D)  7.07

Question 6

Find the integral.
-$\int\left(6 x-6 x^{-1}\right) d x$

Accepted Answer

A)  $12 x^{2}-6 \ln |x|+C$ 
B)  $3 x^{2}-6 \ln |x|+C$ 
C)  $3 x^{2}+3 x^{-2}+C$ 
D)  $12 x^{2}+3 x^{-2}+C$ 
A)  $12 x^{2}-6 \ln |x|+C$ 
B)  $3 x^{2}-6 \ln |x|+C$ 
C)  $3 x^{2}+3 x^{-2}+C$ 
D)  $12 x^{2}+3 x^{-2}+C$

Question 7

Use integration by parts to find the integral.
-$\int \ln 8 x d x$

Accepted Answer

A)  $x \ln 8 x-x+C$ 
B)  $x \ln 8 x-8 x+C$ 
C)  $8 x \ln x-x+C$ 
D)  $x \ln 8 x+x+C$ 
A)  $x \ln 8 x-x+C$ 
B)  $x \ln 8 x-8 x+C$ 
C)  $8 x \ln x-x+C$ 
D)  $x \ln 8 x+x+C$

Question 8

Find the integral.
-$\int 4(2 x+5)^{3} d x$

Accepted Answer

A)  $\frac{3}{4}(2 x+5)^{4}+$ C 
B)  $\frac{1}{2}(2 x+5)^{4}+C$ 
C)  $\frac{3}{8}(2 x+5)^{4}+C$ 
D)  $\frac{1}{4}(2 x+5)^{4}+C$ 
A)  $\frac{3}{4}(2 x+5)^{4}+$ C 
B)  $\frac{1}{2}(2 x+5)^{4}+C$ 
C)  $\frac{3}{8}(2 x+5)^{4}+C$ 
D)  $\frac{1}{4}(2 x+5)^{4}+C$

Question 9

Evaluate the integral.
-$\int_{-1}^{0}\left(3+x^{2}\right) d x$

Accepted Answer

A)  0 
B)  3 
C)  $\frac{10}{3}$ 
D)  -2 
A)  0 
B)  3 
C)  $\frac{10}{3}$ 
D)  -2

Question 10

Use integration by parts to find the integral.
-$\int e^{2 x} x^{2} d x$

Accepted Answer

A)  $\frac{1}{2} x^{2} e^{2 x}-\frac{1}{4} x e^{2} x+\frac{1}{4} e^{2 x}+C$ 
B)  $\frac{1}{2} x^{2} e^{2 x}-\frac{1}{2} x e^{2 x}+\frac{1}{4} e^{2 x}+C$ 
C)  $\frac{1}{2} x^{2} e^{2 x}-\frac{1}{2} x e^{2 x}+C$ 
D)  $\frac{1}{2} x^{2} e^{2 x}-x e^{2 x}+\frac{1}{4} e^{2 x}+C$ 
A)  $\frac{1}{2} x^{2} e^{2 x}-\frac{1}{4} x e^{2} x+\frac{1}{4} e^{2 x}+C$ 
B)  $\frac{1}{2} x^{2} e^{2 x}-\frac{1}{2} x e^{2 x}+\frac{1}{4} e^{2 x}+C$ 
C)  $\frac{1}{2} x^{2} e^{2 x}-\frac{1}{2} x e^{2 x}+C$ 
D)  $\frac{1}{2} x^{2} e^{2 x}-x e^{2 x}+\frac{1}{4} e^{2 x}+C$

Question 11

Use the numeric integration feature on your calculator to find the integral.
-$\int_{-1}^{3}\left(x^{4}+2 x^{3}-2 x^{2}-x-1\right) d x$

Accepted Answer

A)  -10.9333 
B)  13.0667 
C)  10.0667 
D)  6.9333 
A)  -10.9333 
B)  13.0667 
C)  10.0667 
D)  6.9333

Question 12

Find the area bounded by the given curves.
-$y=\frac{1}{2} x^{2}, y=-x^{2}+6$

Accepted Answer

A)  4 
B)  16 
C)  8 
D)  32 
A)  4 
B)  16 
C)  8 
D)  32

Question 13

Solve the problem.
-A force acts on a certain object in such a way that when the object has moved a distance of $r$ (in $m$ ), the force $f$ (in newtons) is given by $f=2 r^{2}+2 r$. Find the work (in joules) done through the first four meters.

Accepted Answer

A)  $64 \mathrm{~J}$ 
B)  $40 \mathrm{~J}$ 
C)  $44.7 \mathrm{~J}$ 
D)  $58.7 \mathrm{~J}$ 
A)  $64 \mathrm{~J}$ 
B)  $40 \mathrm{~J}$ 
C)  $44.7 \mathrm{~J}$ 
D)  $58.7 \mathrm{~J}$

Question 14

Solve the problem.
-Suppose the demand function of a certain item is given by $D(x)=2 e^{x}$ and the supply function is $S(x)=8 e^{-x}$. Find the producer's surplus. Round to three decimal places.

Accepted Answer

A)  0.228 
B)  0.773 
C)  0.664 
D)  1.337 
A)  0.228 
B)  0.773 
C)  0.664 
D)  1.337

Question 15

The given marginal function has x measured in years and MR(x) in hundreds of dollars per year. Use numerical integration on a graphing calculator to find the total revenue over the given period.
-$\operatorname{MR}(x)=0.24 x^{4}-6.25 x^{3}+30.7 x^{2}+87.6 x \quad(0 \leq x \leq 6)$

Accepted Answer

A)  $\$ 2135.45$ 
B)  $\$ 5931.80$ 
C)  $\$ 213,544.80$ 
D)  $\$ 55,864.80$ 
A)  $\$ 2135.45$ 
B)  $\$ 5931.80$ 
C)  $\$ 213,544.80$ 
D)  $\$ 55,864.80$

Question 16

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.
-$f(x)=\frac{8}{x}$ from $x=2$ to $x=10 ; n=4$

Accepted Answer

A)  10.27 
B)  12.60 
C)  16.67 
D)  19.20 
A)  10.27 
B)  12.60 
C)  16.67 
D)  19.20

Question 17

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.
-$f(x)=x^{2}+2$ from $x=1$ to $x=4 ; n=6$

Accepted Answer

A)  30.875 
B)  28.875 
C)  24.875 
D)  26.875 
A)  30.875 
B)  28.875 
C)  24.875 
D)  26.875

Question 18

Evaluate.
-$\int\left(7 x^{2}+x^{-3}\right) d x$

Accepted Answer

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

Question 19

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

Accepted Answer

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

Question 20

Solve the problem.
-For a particular circuit, the current (in amperes) after time $t$ (in seconds) at a certain point $P$ is given by $\mathrm{i}=0.005 \mathrm{t}^{0.27}$. Find the charge (in coulombs) that passes point $\mathrm{P}$ during the first second.

Accepted Answer

A)  $256 \mathrm{C}$ 
B)  $1.27 \mathrm{C}$ 
C)  $0.005 \mathrm{C}$ 
D)  $0.0039 \mathrm{C}$ 
A)  $256 \mathrm{C}$ 
B)  $1.27 \mathrm{C}$ 
C)  $0.005 \mathrm{C}$ 
D)  $0.0039 \mathrm{C}$

Evaluate the definite integral. - $\int_{0}^{3}(4 x+3)^{3} d x$

Use the definite integral to find the area between the $x$ -axis and the graph of $f(x)$ over the indicated interval. - $f(x)=\frac{3}{x^{3}} ;[1,3]$

Find the particular solution of the differential equation. - $\frac{d y}{d x}+y=2 e^{x} ; y=21$ when $x=0$

Solve the problem. -A certain company has found that its expenditure rate per day (in hundreds of dollars) on a certain type of job is given by $E^{\prime}(x)=8 x+7$ , where $x$ is the number of days since the start of the job. Find the expenditure if the job takes 5 days.

Solve the problem. -For a certain drug, the rate of reaction in appropriate units is given by $R^{\prime}(t)=\frac{3}{t}+\frac{2}{t^{2}}$ , where $t$ is measured in hours after the drug is administered. Find the total reaction to the drug from $t=2$ to $\mathrm{t}=8$ .

Find the integral. - $\int\left(6 x-6 x^{-1}\right) d x$

Use integration by parts to find the integral. - $\int \ln 8 x d x$

Find the integral. - $\int 4(2 x+5)^{3} d x$

Evaluate the integral. - $\int_{-1}^{0}\left(3+x^{2}\right) d x$

Use integration by parts to find the integral. - $\int e^{2 x} x^{2} d x$

Use the numeric integration feature on your calculator to find the integral. - $\int_{-1}^{3}\left(x^{4}+2 x^{3}-2 x^{2}-x-1\right) d x$

Find the area bounded by the given curves. - $y=\frac{1}{2} x^{2}, y=-x^{2}+6$

Solve the problem. -A force acts on a certain object in such a way that when the object has moved a distance of $r$ (in $m$ ), the force $f$ (in newtons) is given by $f=2 r^{2}+2 r$ . Find the work (in joules) done through the first four meters.

Solve the problem. -Suppose the demand function of a certain item is given by $D(x)=2 e^{x}$ and the supply function is $S(x)=8 e^{-x}$ . Find the producer's surplus. Round to three decimal places.

The given marginal function has x measured in years and MR(x) in hundreds of dollars per year. Use numerical integration on a graphing calculator to find the total revenue over the given period. - $\operatorname{MR}(x)=0.24 x^{4}-6.25 x^{3}+30.7 x^{2}+87.6 x \quad(0 \leq x \leq 6)$

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. - $f(x)=\frac{8}{x}$ from $x=2$ to $x=10 ; n=4$

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. - $f(x)=x^{2}+2$ from $x=1$ to $x=4 ; n=6$

Evaluate. - $\int\left(7 x^{2}+x^{-3}\right) d x$

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

Solve the problem. -For a particular circuit, the current (in amperes) after time $t$ (in seconds) at a certain point $P$ is given by $\mathrm{i}=0.005 \mathrm{t}^{0.27}$ . Find the charge (in coulombs) that passes point $\mathrm{P}$ during the first second.

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 the definite integral. - ∫03(4x+3)3dx\int_{0}^{3}(4 x+3)^{3} d x∫03​(4x+3)3dx

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)=3x3;[1,3]f(x)=\frac{3}{x^{3}} ;[1,3]f(x)=x33​;[1,3]

Find the particular solution of the differential equation. - dydx+y=2ex;y=21\frac{d y}{d x}+y=2 e^{x} ; y=21dxdy​+y=2ex;y=21 when x=0x=0x=0

Solve the problem. -A certain company has found that its expenditure rate per day (in hundreds of dollars) on a certain type of job is given by E′(x)=8x+7E^{\prime}(x)=8 x+7E′(x)=8x+7 , where xxx is the number of days since the start of the job. Find the expenditure if the job takes 5 days.

Find the integral. - ∫(6x−6x−1)dx\int\left(6 x-6 x^{-1}\right) d x∫(6x−6x−1)dx

Use integration by parts to find the integral. - ∫ln⁡8xdx\int \ln 8 x d x∫ln8xdx

Find the integral. - ∫4(2x+5)3dx\int 4(2 x+5)^{3} d x∫4(2x+5)3dx

Evaluate the integral. - ∫−10(3+x2)dx\int_{-1}^{0}\left(3+x^{2}\right) d x∫−10​(3+x2)dx

Use integration by parts to find the integral. - ∫e2xx2dx\int e^{2 x} x^{2} d x∫e2xx2dx

Use the numeric integration feature on your calculator to find the integral. - ∫−13(x4+2x3−2x2−x−1)dx\int_{-1}^{3}\left(x^{4}+2 x^{3}-2 x^{2}-x-1\right) d x∫−13​(x4+2x3−2x2−x−1)dx

Find the area bounded by the given curves. - y=12x2,y=−x2+6y=\frac{1}{2} x^{2}, y=-x^{2}+6y=21​x2,y=−x2+6

Solve the problem. -A force acts on a certain object in such a way that when the object has moved a distance of rrr (in mmm ), the force fff (in newtons) is given by f=2r2+2rf=2 r^{2}+2 rf=2r2+2r . Find the work (in joules) done through the first four meters.

Solve the problem. -Suppose the demand function of a certain item is given by D(x)=2exD(x)=2 e^{x}D(x)=2ex and the supply function is S(x)=8e−xS(x)=8 e^{-x}S(x)=8e−x . Find the producer's surplus. Round to three decimal places.

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)=8xf(x)=\frac{8}{x}f(x)=x8​ from x=2x=2x=2 to x=10;n=4x=10 ; n=4x=10;n=4

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)=x2+2f(x)=x^{2}+2f(x)=x2+2 from x=1x=1x=1 to x=4;n=6x=4 ; n=6x=4;n=6

Evaluate. - ∫(7x2+x−3)dx\int\left(7 x^{2}+x^{-3}\right) d x∫(7x2+x−3)dx

Find the integral. - ∫exdxex+e\int \frac{e^{x} d x}{e^{x}+e}∫ex+eexdx​

Solve the problem. -For a particular circuit, the current (in amperes) after time ttt (in seconds) at a certain point PPP is given by i=0.005t0.27\mathrm{i}=0.005 \mathrm{t}^{0.27}i=0.005t0.27 . Find the charge (in coulombs) that passes point P\mathrm{P}P during the first second.