Question 1

Find the area of the region that lies beneath the given curve.  y=sin⁡x,0≤x≤π3y = \sin x , 0 \leq x \leq \frac { \pi } { 3 }y=sinx,0≤x≤3π​

Accepted Answer

A)    1.5001.5001.500 
B)    1.4501.4501.450 
C)    −0.500- 0.500−0.500 
D)    −1.500- 1.500−1.500 
E)    0.5000.5000.500

Question 2

Find the indefinite integral.  ∫x−2/3x1/3−2dx\int x ^ { - 2 / 3 } \sqrt { x ^ { 1 / 3 } - 2 } d x∫x−2/3x1/3−2​dx

Accepted Answer

A)    2(x1/3−2) 3/2+C2 \left( x ^ { 1 / 3 } - 2 \right)  ^ { 3 / 2 } + C2(x1/3−2) 3/2+C 
B)    2x1/3(x1/3−2) 3/2+C2 x ^ { 1 / 3 } \left( x ^ { 1 / 3 } - 2 \right)  ^ { 3 / 2 } + C2x1/3(x1/3−2) 3/2+C 
C)    12(x1/3−2) 3/2+C\frac { 1 } { 2 } \left( x ^ { 1 / 3 } - 2 \right)  ^ { 3 / 2 } + C21​(x1/3−2) 3/2+C 
D)    x1/3(x1/3−2) 3/2+Cx ^ { 1 / 3 } \left( x ^ { 1 / 3 } - 2 \right)  ^ { 3 / 2 } + Cx1/3(x1/3−2) 3/2+C

Question 3

Find the derivative of the function. $$g ( x ) = \int _ { 4 } ^ { x ^ { 2 } } 4 \sqrt { 1 + t ^ { 2 } } d t$$

Accepted Answer

The answer of Find the derivative of the function. \[g...

Question 4

The graph of a function $f$ on the interval $[ 0,8 ]$ is shown in the figure. Compute the Riemann sum for $f$ on $[ 0,8 ]$ using four subintervals of equal length and choosing the evaluation points to be (a) the left endpoints, (b) the right endpoints, and (c) the midpoints of the subintervals.

Accepted Answer

The answer of The graph of a function $f$ on...

Question 5

Find the indefinite integral.  ∫(x2+4x−5) 2(2x+4) dx\int \left( x ^ { 2 } + 4 x - 5 \right)  ^ { 2 } ( 2 x + 4 )  d x∫(x2+4x−5) 2(2x+4) dx

Accepted Answer

A)    −2(x2+4x−5) 3+C- 2 \left( x ^ { 2 } + 4 x - 5 \right)  ^ { 3 } + C−2(x2+4x−5) 3+C 
B)    2(x2+4x−5) 3+C2 \left( x ^ { 2 } + 4 x - 5 \right)  ^ { 3 } + C2(x2+4x−5) 3+C 
C)    (x2+4x−5) 3+C\left( x ^ { 2 } + 4 x - 5 \right)  ^ { 3 } + C(x2+4x−5) 3+C 
D)    13(x2+4x−5) 3+C\frac { 1 } { 3 } \left( x ^ { 2 } + 4 x - 5 \right)  ^ { 3 } + C31​(x2+4x−5) 3+C

Question 6

Approximate the area under the curve $y = \frac { 5 } { x ^ { 2 } }$ from 1 to 2 using ten approximating rectangles of equal widths and right endpoints. Round the answer to the nearest hundredth.

Accepted Answer

The answer of Approximate the area under the curve \(y...

Question 7

If  hth ^ { t }ht  is a child's rate of growth in pounds per year, which of the following expressions represents the increase in the child's weight (in pounds)  between the years 3 and 7 ?

Accepted Answer

A)    ∫37ht(t) dt\int _ { 3 } ^ { 7 } h ^ { t } ( t )  d t∫37​ht(t) dt 
B)    ht(7) −ht(3) h ^ { t} ( 7 )  - h ^ { t } ( 3 ) ht(7) −ht(3)  
C)    ∫73h(t) dt\int _ { 7 } ^ { 3 } h ( t )  d t∫73​h(t) dt

Question 8

Evaluate the indefinite integral. $$\int \sec ^ { 2 } x \tan x d x$$

Accepted Answer

The answer of Evaluate the indefinite integral. \[\int \sec ^...

Question 9

Find the integral using an appropriate trigonometric substitution.  ∫x4−x2dx\int \frac { x } { \sqrt { 4 - x ^ { 2 } } } d x∫4−x2​x​dx

Accepted Answer

A)    −2−x+C- \sqrt { 2 - x } + C−2−xc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​+C 
B)    2−x+C\sqrt { 2 - x } + C2−xc-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​+C 
C)    4−x2+C\sqrt { 4 - x ^ { 2 } } + C4−x2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​+C 
D)    −4−x2+C- \sqrt { 4 - x ^ { 2 } } + C−4−x2c-2.7,0,-7.17,-2.7,-13.5,-8c-5.8,-5.3,-9.5,-10,-9.5,-14c0,-2,0.3,-3.3,1,-4c1.3,-2.7,23.83,-20.7,67.5,-54c44.2,-33.3,65.8,-50.3,66.5,-51c1.3,-1.3,3,-2,5,-2c4.7,0,8.7,3.3,12,10s173,378,173,378c0.7,0,35.3,-71,104,-213c68.7,-142,137.5,-285,206.5,-429c69,-144,104.5,-217.7,106.5,-221l0 -0c5.3,-9.3,12,-14,20,-14H400000v40H845.2724s-225.272,467,-225.272,467s-235,486,-235,486c-2.7,4.7,-9,7,-19,7c-6,0,-10,-1,-12,-3s-194,-422,-194,-422s-65,47,-65,47zM834 80h400000v40h-400000z'/>​+C

Question 10

$$\text { Find a function } f ( x ) \text { such that } 5 + \int _ { a } ^ { x } \frac { f ( t ) } { t ^ { 2 } } d t = 12 \sqrt { x } \text { for } x > 0 \text { and some number } a \text {. }$$

Accepted Answer

The answer of \[\text { Find a function } f...

Question 11

$$\text { Find the interval on which the curve } F ( x ) = \int _ { 0 } ^ { x } \frac { d t } { 11 + 11 t } \text { is concave downward. }$$

Accepted Answer

The answer of \[\text { Find the interval on which...

Question 12

Find the derivative of the function. $$g ( x ) = \int _ { \cos x } ^ { 9 x } \cos \left( t ^ { 3 } \right) d t$$

Accepted Answer

The answer of Find the derivative of the function. \[g...

Question 13

The table gives the values of a function obtained from an experiment. Use the values to estimate $\int _ { 0 } ^ { 6 } f ( w ) d w$ using three equal subintervals with left endpoints.
$\begin{array}{|c|c|c|c|c|c|c|c|}
\hline w & 0 & 1 & 2 & 3 & 4 & 5 & 6 \
\hline f(w) & 9.7 & 9.1 & 7.7 & 6.1 & 4.2 & -6.6 & -10.3 \
\hline
\end{array}$

Accepted Answer

The answer of The table gives the values of a...

Question 14

Evaluate the integral if it exists.  ∫(5−xx) 2dx\int \left( \frac { 5 - x } { x } \right)  ^ { 2 } d x∫(x5−x​) 2dx

Accepted Answer

A)    1−10ln⁡x+C1 - 10 \ln x + C1−10lnx+C 
B)   x−10ln⁡x−25x+Cx - 10 \ln x - \frac { 25 } { x } + Cx−10lnx−x25​+C 
C)    ln⁡x−25x+C\ln x - 25 x + Clnx−25x+C 
D)   x−110ln⁡x+Cx - \frac { 1 } { 10 \ln x } + Cx−10lnx1​+C 
E)   none of these

Question 15

Find the area of the region under the graph of  fff  on  [a,b][ a , b ][a,b] .
 f(x) =x2−2x+2;[−1,2]f ( x )  = x ^ { 2 } - 2 x + 2 ; \quad [ - 1,2 ]f(x) =x2−2x+2;[−1,2]

Accepted Answer

A)    −3- 3−3 
B)    −6- 6−6 
C)   6d 3

Question 16

Find the indefinite integral.
 ∫(5−8x3+3x6) dx\int \left( 5 - 8 x ^ { 3 } + 3 x ^ { 6 } \right)  d x∫(5−8x3+3x6) dx

Accepted Answer

A)    37x7−2x4+5x+C\frac { 3 } { 7 } x ^ { 7 } - 2 x ^ { 4 } + 5 x + C73​x7−2x4+5x+C 
B)    5x−8x4+3x7+C5 x - 8 x ^ { 4 } + 3 x ^ { 7 } + C5x−8x4+3x7+C 
C)    37x6−2x3+5x+C\frac { 3 } { 7 } x ^ { 6 } - 2 x ^ { 3 } + 5 x + C73​x6−2x3+5x+C 
D)    5x−24x2+18x5+C5 x - 24 x ^ { 2 } + 18 x ^ { 5 } + C5x−24x2+18x5+C

Question 17

Find the indefinite integral.
 ∫2x9−6x2+3x9dx\int \frac { 2 x ^ { 9 } - 6 x ^ { 2 } + 3 } { x ^ { 9 } } d x∫x92x9−6x2+3​dx

Accepted Answer

A)    2x+1x6−38x8+C2 x + \frac { 1 } { x ^ { 6 } } - \frac { 3 } { 8 x ^ { 8 } } + C2x+x61​−8x83​+C 
B)   2x+34x8−310x10+C2 x + \frac { 3 } { 4 x ^ { 8 } } - \frac { 3 } { 10 x ^ { 10 } } + C2x+4x83​−10x103​+C 
C)   2x+x6−38x8+C2 x + x ^ { 6 } - \frac { 3 } { 8 } x ^ { 8 } + C2x+x6−83​x8+C 
D)    2x+34x8−310x10+C2 x + \frac { 3 } { 4 } x ^ { 8 } - \frac { 3 } { 10 } x ^ { 10 } + C2x+43​x8−103​x10+C

Question 18

Find the general indefinite integral. $$\int \frac { \sin 18 t } { \sin 9 t } d t$$

Accepted Answer

The answer of Find the general indefinite integral. \[\int \frac...

Question 19

Evaluate the integral.
 ∫x/3x/7sin⁡tdt\int _ { x / 3 } ^ { x / 7 } \sin t d t∫x/3x/7​sintdt

Accepted Answer

A)    −0.500- 0.500−0.500 
B)    −1.000- 1.000−1.000 
C)    −0.400- 0.400−0.400 
D)    0.2500.2500.250 
E)    1.0001.0001.000

Question 20

Evaluate the integral by making the given substitution. $$\int \cos 3 x d x , u = 3 x$$

Accepted Answer

The answer of Evaluate the integral by making the given...

Exam 4: Applications of Differentiation

Find the indefinite integral. $\int \left( 5 - 8 x ^ { 3 } + 3 x ^ { 6 } \right) d x$

Correct Answer
verified

The graph of a function $f$ on the interval $[ 0,8 ]$ is shown in the figure. Compute the Riemann sum for $f$ on $[ 0,8 ]$ using four subintervals of equal length and choosing the evaluation points to be (a) the left endpoints, (b) the right endpoints, and (c) the midpoints of the subintervals.

Correct Answer
verified

Find the integral using an appropriate trigonometric substitution. $\int \frac { x } { \sqrt { 4 - x ^ { 2 } } } d x$

Correct Answer
verified

Correct Answer
verified

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

Correct Answer
verified

Find the indefinite integral. $\int x ^ { - 2 / 3 } \sqrt { x ^ { 1 / 3 } - 2 } d x$

Correct Answer
verified

Find the general indefinite integral. $\int \frac { \sin 18 t } { \sin 9 t } d t$

Correct Answer
verified

Find the indefinite integral. $\int \frac { 2 x ^ { 9 } - 6 x ^ { 2 } + 3 } { x ^ { 9 } } d x$

Correct Answer
verified

Find the area of the region under the graph of $f$ on $[ a , b ]$ . $f ( x ) = x ^ { 2 } - 2 x + 2 ; \quad [ - 1,2 ]$

Correct Answer
verified

$\text { Find a function } f ( x ) \text { such that } 5 + \int _ { a } ^ { x } \frac { f ( t ) } { t ^ { 2 } } d t = 12 \sqrt { x } \text { for } x > 0 \text { and some number } a \text {. }$

Correct Answer
verified

$\text { Find the interval on which the curve } F ( x ) = \int _ { 0 } ^ { x } \frac { d t } { 11 + 11 t } \text { is concave downward. }$

Correct Answer
verified

Evaluate the indefinite integral. $\int \sec ^ { 2 } x \tan x d x$

Correct Answer
verified

Find the derivative of the function. $g ( x ) = \int _ { 4 } ^ { x ^ { 2 } } 4 \sqrt { 1 + t ^ { 2 } } d t$

Correct Answer
verified

Find the area of the region that lies beneath the given curve. $y = \sin x , 0 \leq x \leq \frac { \pi } { 3 }$

Correct Answer
verified

Evaluate the integral if it exists. $\int \left( \frac { 5 - x } { x } \right) ^ { 2 } d x$

Correct Answer
verified

Evaluate the integral by making the given substitution. $\int \cos 3 x d x , u = 3 x$

Correct Answer
verified

If $h ^ { t }$ is a child's rate of growth in pounds per year, which of the following expressions represents the increase in the child's weight (in pounds) between the years 3 and 7 ?

Correct Answer
verified

Evaluate the integral. $\int _ { x / 3 } ^ { x / 7 } \sin t d t$

Correct Answer
verified

Approximate the area under the curve $y = \frac { 5 } { x ^ { 2 } }$ from 1 to 2 using ten approximating rectangles of equal widths and right endpoints. Round the answer to the nearest hundredth.

Correct Answer
verified

Find the derivative of the function. $g ( x ) = \int _ { \cos x } ^ { 9 x } \cos \left( t ^ { 3 } \right) d t$

Correct Answer
verified

Exam 4: Applications of Differentiation

Find the indefinite integral. ∫(5−8x3+3x6) dx\int \left( 5 - 8 x ^ { 3 } + 3 x ^ { 6 } \right) d x∫(5−8x3+3x6) dx

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Find the integral using an appropriate trigonometric substitution. ∫x4−x2dx\int \frac { x } { \sqrt { 4 - x ^ { 2 } } } d x∫4−x2​x​dx

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Find the indefinite integral. ∫(x2+4x−5) 2(2x+4) dx\int \left( x ^ { 2 } + 4 x - 5 \right) ^ { 2 } ( 2 x + 4 ) d x∫(x2+4x−5) 2(2x+4) dx

Correct AnswerverifiedShow Answer

Find the indefinite integral. ∫x−2/3x1/3−2dx\int x ^ { - 2 / 3 } \sqrt { x ^ { 1 / 3 } - 2 } d x∫x−2/3x1/3−2​dx

Correct AnswerverifiedShow Answer

Find the general indefinite integral. ∫sin⁡18tsin⁡9tdt\int \frac { \sin 18 t } { \sin 9 t } d t∫sin9tsin18t​dt

Correct AnswerverifiedShow Answer

Find the indefinite integral. ∫2x9−6x2+3x9dx\int \frac { 2 x ^ { 9 } - 6 x ^ { 2 } + 3 } { x ^ { 9 } } d x∫x92x9−6x2+3​dx

Correct AnswerverifiedShow Answer

Find the area of the region under the graph of fff on [a,b][ a , b ][a,b] . f(x) =x2−2x+2;[−1,2]f ( x ) = x ^ { 2 } - 2 x + 2 ; \quad [ - 1,2 ]f(x) =x2−2x+2;[−1,2]

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

Find the interval on which the curve F(x)=∫0xdt11+11t is concave downward. \text { Find the interval on which the curve } F ( x ) = \int _ { 0 } ^ { x } \frac { d t } { 11 + 11 t } \text { is concave downward. } Find the interval on which the curve F(x)=∫0x​11+11tdt​ is concave downward.

Correct AnswerverifiedShow Answer

Evaluate the indefinite integral. ∫sec⁡2xtan⁡xdx\int \sec ^ { 2 } x \tan x d x∫sec2xtanxdx

Correct AnswerverifiedShow Answer

Find the derivative of the function. g(x)=∫4x241+t2dtg ( x ) = \int _ { 4 } ^ { x ^ { 2 } } 4 \sqrt { 1 + t ^ { 2 } } d tg(x)=∫4x2​41+t2​dt

Correct AnswerverifiedShow Answer

Find the area of the region that lies beneath the given curve. y=sin⁡x,0≤x≤π3y = \sin x , 0 \leq x \leq \frac { \pi } { 3 }y=sinx,0≤x≤3π​

Correct AnswerverifiedShow Answer

Evaluate the integral if it exists. ∫(5−xx) 2dx\int \left( \frac { 5 - x } { x } \right) ^ { 2 } d x∫(x5−x​) 2dx

Correct AnswerverifiedShow Answer

Evaluate the integral by making the given substitution. ∫cos⁡3xdx,u=3x\int \cos 3 x d x , u = 3 x∫cos3xdx,u=3x

Correct AnswerverifiedShow Answer

If hth ^ { t }ht is a child's rate of growth in pounds per year, which of the following expressions represents the increase in the child's weight (in pounds) between the years 3 and 7 ?

Correct AnswerverifiedShow Answer

Evaluate the integral. ∫x/3x/7sin⁡tdt\int _ { x / 3 } ^ { x / 7 } \sin t d t∫x/3x/7​sintdt

Correct AnswerverifiedShow Answer

Approximate the area under the curve y=5x2y = \frac { 5 } { x ^ { 2 } }y=x25​ from 1 to 2 using ten approximating rectangles of equal widths and right endpoints. Round the answer to the nearest hundredth.

Correct AnswerverifiedShow Answer

Find the derivative of the function. g(x)=∫cos⁡x9xcos⁡(t3)dtg ( x ) = \int _ { \cos x } ^ { 9 x } \cos \left( t ^ { 3 } \right) d tg(x)=∫cosx9x​cos(t3)dt

Correct AnswerverifiedShow Answer

Find the indefinite integral. $\int \left( 5 - 8 x ^ { 3 } + 3 x ^ { 6 } \right) d x$

Correct Answer
verified

Correct Answer
verified

Find the integral using an appropriate trigonometric substitution. $\int \frac { x } { \sqrt { 4 - x ^ { 2 } } } d x$

Correct Answer
verified

Correct Answer
verified

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

Correct Answer
verified

Find the indefinite integral. $\int x ^ { - 2 / 3 } \sqrt { x ^ { 1 / 3 } - 2 } d x$

Correct Answer
verified

Find the general indefinite integral. $\int \frac { \sin 18 t } { \sin 9 t } d t$

Correct Answer
verified

Find the indefinite integral. $\int \frac { 2 x ^ { 9 } - 6 x ^ { 2 } + 3 } { x ^ { 9 } } d x$

Correct Answer
verified

Find the area of the region under the graph of $f$ on $[ a , b ]$ . $f ( x ) = x ^ { 2 } - 2 x + 2 ; \quad [ - 1,2 ]$

Correct Answer
verified

Correct Answer
verified

$\text { Find the interval on which the curve } F ( x ) = \int _ { 0 } ^ { x } \frac { d t } { 11 + 11 t } \text { is concave downward. }$

Correct Answer
verified

Evaluate the indefinite integral. $\int \sec ^ { 2 } x \tan x d x$

Correct Answer
verified

Find the derivative of the function. $g ( x ) = \int _ { 4 } ^ { x ^ { 2 } } 4 \sqrt { 1 + t ^ { 2 } } d t$

Correct Answer
verified

Find the area of the region that lies beneath the given curve. $y = \sin x , 0 \leq x \leq \frac { \pi } { 3 }$

Correct Answer
verified

Evaluate the integral if it exists. $\int \left( \frac { 5 - x } { x } \right) ^ { 2 } d x$

Correct Answer
verified

Evaluate the integral by making the given substitution. $\int \cos 3 x d x , u = 3 x$

Correct Answer
verified

If $h ^ { t }$ is a child's rate of growth in pounds per year, which of the following expressions represents the increase in the child's weight (in pounds) between the years 3 and 7 ?

Correct Answer
verified

Evaluate the integral. $\int _ { x / 3 } ^ { x / 7 } \sin t d t$

Correct Answer
verified

Approximate the area under the curve $y = \frac { 5 } { x ^ { 2 } }$ from 1 to 2 using ten approximating rectangles of equal widths and right endpoints. Round the answer to the nearest hundredth.

Correct Answer
verified

Find the derivative of the function. $g ( x ) = \int _ { \cos x } ^ { 9 x } \cos \left( t ^ { 3 } \right) d t$

Correct Answer
verified