Question 1

Evaluate.
-$\int\left(7 x^{2}+1\right) d x$

Accepted Answer

A)  $\frac{7}{3} x^{3}+x+C$ 
B)  $\frac{7}{3} x^{3}+C$ 
C)  $14 x+C$ 
D)  $x+C$ 
A)  $\frac{7}{3} x^{3}+x+C$ 
B)  $\frac{7}{3} x^{3}+C$ 
C)  $14 x+C$ 
D)  $x+C$

Question 2

Solve the problem.
-A certain object moves in such a way that its velocity (in $\mathrm{m} / \mathrm{s}$ ) after time $\mathrm{t}$ (in $\mathrm{s}$ ) is given by $v=t^{2}+2 t+5$. Find the distance traveled during the first four seconds.

Accepted Answer

A)  $57.3 \mathrm{~m}$ 
B)  $37.3 \mathrm{~m}$ 
C)  $36.0 \mathrm{~m}$ 
D)  $29.0 \mathrm{~m}$ 
A)  $57.3 \mathrm{~m}$ 
B)  $37.3 \mathrm{~m}$ 
C)  $36.0 \mathrm{~m}$ 
D)  $29.0 \mathrm{~m}$

Question 3

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 consumer's surplus. Round to three decimal places.

Accepted Answer

A)  1.227 
B)  0.773 
C)  1.337 
D)  0.664 
A)  1.227 
B)  0.773 
C)  1.337 
D)  0.664

Question 4

Use the definite integral to find the area between the $x$-axis and the graph of $f(x)$ over the indicated interval.
-$f(x)=e^{x}-1 ;[-2,3]$

Accepted Answer

A)  7.7 
B)  17.2 
C)  17.7 
D)  15.0 
A)  7.7 
B)  17.2 
C)  17.7 
D)  15.0

Question 5

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.
-$f(x)=e^{-x}+3$ from $x=-2$ to $x=6 ; n=4$

Accepted Answer

A)  26.31 
B)  30.29 
C)  41.09 
D)  24.02 
A)  26.31 
B)  30.29 
C)  41.09 
D)  24.02

Question 6

Evaluate the definite integral.
-$\int^{\frac{1}{9}} e^{9 x} d x$

Accepted Answer

A)  $\frac{1}{9} \mathrm{e}-\frac{1}{9}$ 
B)  $9 e-9$ 
C)  $\frac{1}{9} \mathrm{e}$ 
D)  $9 \mathrm{e}$ 
A)  $\frac{1}{9} \mathrm{e}-\frac{1}{9}$ 
B)  $9 e-9$ 
C)  $\frac{1}{9} \mathrm{e}$ 
D)  $9 \mathrm{e}$

Question 7

Evaluate the definite integral.
-$\int_{2}^{3} \frac{x}{\sqrt{2 x^{2}+4}} d x$

Accepted Answer

A)  2.3452 
B)  0.6132 
C)  2.4526 
D)  9.8105 
A)  2.3452 
B)  0.6132 
C)  2.4526 
D)  9.8105

Question 8

Provide an appropriate response.
-Suppose we wish to use a substitution to find $\left(x^{2}+2\right) d x$, and we let $u=x^{2}+2$. Which of the following would be the correct results?
i. $\int u d u$
ii. $\int\left(\frac{u}{2}\right) d u$
iii. $\int(u-2) d u$

Accepted Answer

A)  Both i and iii 
B)  Only i 
C)  Only ii 
D)  None of these is correct. 
A)  Both i and iii 
B)  Only i 
C)  Only ii 
D)  None of these is correct.

Question 9

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

Accepted Answer

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

Question 10

Evaluate.
-$\int 8 e^{4 y} d y$

Accepted Answer

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

Question 11

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

Accepted Answer

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

Question 12

Find the particular solution of the differential equation.
-$\frac{d y}{d x}+2 x=3 x^{2} ; y=24$ when $x=0$

Accepted Answer

A)  $y=x^{3}+2 x^{2}+24$ 
B)  $y=3 x^{3}+2 x^{2}+24$ 
C)  $y=3 x^{3}+x^{2}+24$ 
D)  $y=x^{3}-x^{2}+24$ 
A)  $y=x^{3}+2 x^{2}+24$ 
B)  $y=3 x^{3}+2 x^{2}+24$ 
C)  $y=3 x^{3}+x^{2}+24$ 
D)  $y=x^{3}-x^{2}+24$

Question 13

Find the particular solution of the differential equation.
-$x \frac{d y}{d x}+5 y=x^{2} ; y=12$ when $x=2$

Accepted Answer

A)  $y=\frac{8}{x^{2}}-\frac{3}{x^{3}}$ 
B)  $y=\frac{x^{2}}{7}+\frac{2560}{7 x^{5}}$ 
C)  $y=\frac{x^{2}}{7}+\frac{2560}{x^{5}}$ 
D)  $y=\frac{7}{x^{2}}+7 x^{5}$ 
A)  $y=\frac{8}{x^{2}}-\frac{3}{x^{3}}$ 
B)  $y=\frac{x^{2}}{7}+\frac{2560}{7 x^{5}}$ 
C)  $y=\frac{x^{2}}{7}+\frac{2560}{x^{5}}$ 
D)  $y=\frac{7}{x^{2}}+7 x^{5}$

Question 14

Find the integral.
-$\int \frac{x}{(3 x+1)^{1 / 3}} d x$

Accepted Answer

A)  $\frac{1}{3}\left[\frac{(3 x+1)^{5 / 3}}{5}-\frac{(3 x+1)^{2 / 3}}{2}\right]+C$ 
B)  $\frac{(3 x+1)^{5 / 3}}{5}-\frac{(3 x+1)^{2 / 3}}{2}+C$ 
C)  $\frac{1}{3} \frac{(3 x+1)^{5 / 3}}{5}-\frac{(3 x+1)^{2 / 3}}{2}+C$ 
D)  $\frac{1}{3}\left[\frac{(3 x+1)^{5 / 3}}{2}-\frac{(3 x+1)^{2 / 3}}{5}\right]+C$ 
A)  $\frac{1}{3}\left[\frac{(3 x+1)^{5 / 3}}{5}-\frac{(3 x+1)^{2 / 3}}{2}\right]+C$ 
B)  $\frac{(3 x+1)^{5 / 3}}{5}-\frac{(3 x+1)^{2 / 3}}{2}+C$ 
C)  $\frac{1}{3} \frac{(3 x+1)^{5 / 3}}{5}-\frac{(3 x+1)^{2 / 3}}{2}+C$ 
D)  $\frac{1}{3}\left[\frac{(3 x+1)^{5 / 3}}{2}-\frac{(3 x+1)^{2 / 3}}{5}\right]+C$

Question 15

Find the general solution of the differential equation.
-$\frac{d y}{d x}-15 x^{2}=8$

Accepted Answer

A)  $5 x^{3}+4 x+C$ 
B)  $5 x^{3}+8 x+C$ 
C)  $5 x^{3}-4 x+C$ 
D)  $-5 x^{3}+8 x+C$ 
A)  $5 x^{3}+4 x+C$ 
B)  $5 x^{3}+8 x+C$ 
C)  $5 x^{3}-4 x+C$ 
D)  $-5 x^{3}+8 x+C$

Question 16

Solve the problem.
-Find $C(x)$ if $C^{\prime}(x)=\sqrt{x}$ and $C(9)=40$.

Accepted Answer

A)  $C(x)=\frac{2}{3} x^{3 / 2}+31$ 
B)  $C(x)=2 x^{3 / 2}+22$ 
C)  $C(x)=2 x^{3 / 2}+31$ 
D)  $C(x)=\frac{2}{3} x^{3 / 2}+22$ 
A)  $C(x)=\frac{2}{3} x^{3 / 2}+31$ 
B)  $C(x)=2 x^{3 / 2}+22$ 
C)  $C(x)=2 x^{3 / 2}+31$ 
D)  $C(x)=\frac{2}{3} x^{3 / 2}+22$

Question 17

Find the area of the shaded region.
-

Accepted Answer

A)  $\frac{23}{3}$ 
B)  $\frac{25}{3}$ 
C)  $\frac{26}{3}$ 
D)  $\frac{22}{3}$ 
A)  $\frac{23}{3}$ 
B)  $\frac{25}{3}$ 
C)  $\frac{26}{3}$ 
D)  $\frac{22}{3}$

Question 18

Find the integral.
-$\int \frac{\mathrm{dr}}{\sqrt{6 \mathrm{r}-7}}$

Accepted Answer

A)  $\frac{1}{3} \sqrt{6 r-7}+C$ 
B)  $\frac{1}{4} \sqrt{6 r-7}+C$ 
C)  $\frac{1}{6} \sqrt{6 r-7}+C$ 
D)  $\frac{1}{2} \sqrt{6 r-7}+C$ 
A)  $\frac{1}{3} \sqrt{6 r-7}+C$ 
B)  $\frac{1}{4} \sqrt{6 r-7}+C$ 
C)  $\frac{1}{6} \sqrt{6 r-7}+C$ 
D)  $\frac{1}{2} \sqrt{6 r-7}+C$

Question 19

Use the definite integral to find the area between the $x$-axis and the graph of $f(x)$ over the indicated interval.
-$f(x)=-x^{2}+9 ;[0,5]$

Accepted Answer

A)  $\frac{10}{9}$ 
B)  $\frac{98}{3}$ 
C)  $\frac{10}{3}$ 
D)  $\frac{5}{9}$ 
A)  $\frac{10}{9}$ 
B)  $\frac{98}{3}$ 
C)  $\frac{10}{3}$ 
D)  $\frac{5}{9}$

Question 20

Use the numeric integration feature on your calculator to find the integral.
-$\int_{1}^{3} \ln (t) d t$

Accepted Answer

A)  None of the above is correct. 
B)  2.29584 
C)  1.29584 
D)  0.29584 
A)  None of the above is correct. 
B)  2.29584 
C)  1.29584 
D)  0.29584

Evaluate. - $\int\left(7 x^{2}+1\right) d x$

Solve the problem. -A certain object moves in such a way that its velocity (in $\mathrm{m} / \mathrm{s}$ ) after time $\mathrm{t}$ (in $\mathrm{s}$ ) is given by $v=t^{2}+2 t+5$ . Find the distance traveled during the first four seconds.

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 consumer's surplus. Round to three decimal places.

Use the definite integral to find the area between the $x$ -axis and the graph of $f(x)$ over the indicated interval. - $f(x)=e^{x}-1 ;[-2,3]$

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. - $f(x)=e^{-x}+3$ from $x=-2$ to $x=6 ; n=4$

Evaluate the definite integral. - $\int^{\frac{1}{9}} e^{9 x} d x$

Evaluate the definite integral. - $\int_{2}^{3} \frac{x}{\sqrt{2 x^{2}+4}} d x$

Provide an appropriate response. -Suppose we wish to use a substitution to find $\left(x^{2}+2\right) d x$ , and we let $u=x^{2}+2$ . Which of the following would be the correct results? i. $\int u d u$ ii. $\int\left(\frac{u}{2}\right) d u$ iii. $\int(u-2) d u$

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

Evaluate. - $\int 8 e^{4 y} d y$

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

Find the particular solution of the differential equation. - $\frac{d y}{d x}+2 x=3 x^{2} ; y=24$ when $x=0$

Find the particular solution of the differential equation. - $x \frac{d y}{d x}+5 y=x^{2} ; y=12$ when $x=2$

Find the integral. - $\int \frac{x}{(3 x+1)^{1 / 3}} d x$

Find the general solution of the differential equation. - $\frac{d y}{d x}-15 x^{2}=8$

Solve the problem. -Find $C(x)$ if $C^{\prime}(x)=\sqrt{x}$ and $C(9)=40$ .

Find the area of the shaded region. -

Find the integral. - $\int \frac{\mathrm{dr}}{\sqrt{6 \mathrm{r}-7}}$

Use the definite integral to find the area between the $x$ -axis and the graph of $f(x)$ over the indicated interval. - $f(x)=-x^{2}+9 ;[0,5]$

Use the numeric integration feature on your calculator to find the integral. - $\int_{1}^{3} \ln (t) d t$

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Exam 13: Integral Calculus

Evaluate. - ∫(7x2+1)dx\int\left(7 x^{2}+1\right) d x∫(7x2+1)dx

Solve the problem. -A certain object moves in such a way that its velocity (in m/s\mathrm{m} / \mathrm{s}m/s ) after time t\mathrm{t}t (in s\mathrm{s}s ) is given by v=t2+2t+5v=t^{2}+2 t+5v=t2+2t+5 . Find the distance traveled during the first four seconds.

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 consumer's surplus. Round to three decimal places.

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)=ex−1;[−2,3]f(x)=e^{x}-1 ;[-2,3]f(x)=ex−1;[−2,3]

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)=e−x+3f(x)=e^{-x}+3f(x)=e−x+3 from x=−2x=-2x=−2 to x=6;n=4x=6 ; n=4x=6;n=4

Evaluate the definite integral. - ∫19e9xdx\int^{\frac{1}{9}} e^{9 x} d x∫91​e9xdx

Evaluate the definite integral. - ∫23x2x2+4dx\int_{2}^{3} \frac{x}{\sqrt{2 x^{2}+4}} d x∫23​2x2+4​x​dx

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

Evaluate. - ∫8e4ydy\int 8 e^{4 y} d y∫8e4ydy

Evaluate the integral. - ∫−11(x2+1)dx\int_{-1}^{1}\left(x^{2}+1\right) d x∫−11​(x2+1)dx

Find the particular solution of the differential equation. - dydx+2x=3x2;y=24\frac{d y}{d x}+2 x=3 x^{2} ; y=24dxdy​+2x=3x2;y=24 when x=0x=0x=0

Find the particular solution of the differential equation. - xdydx+5y=x2;y=12x \frac{d y}{d x}+5 y=x^{2} ; y=12xdxdy​+5y=x2;y=12 when x=2x=2x=2

Find the integral. - ∫x(3x+1)1/3dx\int \frac{x}{(3 x+1)^{1 / 3}} d x∫(3x+1)1/3x​dx

Find the general solution of the differential equation. - dydx−15x2=8\frac{d y}{d x}-15 x^{2}=8dxdy​−15x2=8

Solve the problem. -Find C(x)C(x)C(x) if C′(x)=xC^{\prime}(x)=\sqrt{x}C′(x)=x​ and C(9)=40C(9)=40C(9)=40 .