Question 1

Provide an appropriate response.
-If we use  u=x2u=x^{2}u=x2  as a substitution to find  xex2dxx e^{x^{2}} d xxex2dx , then which of the following would be the correct results?
i.  ∫(eu2) du\int\left(\frac{e^{u}}{2}\right)  d u \quad∫(2eu​) du  ii)   ∫(2eu) du\int\left(2 e^{u}\right)  d u \quad∫(2eu) du  iii)   ∫(ueu) du\int\left(u e^{u}\right)  d u∫(ueu) du

Accepted Answer

A)   i is correct.
B)   iii is correct.
C)   ii is correct.
D)   None of these is correct. E) A) and D)
F) B) and C)

Question 2

Find the integral.
- ∫(x6+6x) dx\int\left(\frac{x}{6}+\frac{6}{x}\right)  d x∫(6x​+x6​) dx

Accepted Answer

A)    112x2+6ln⁡∣x∣+C\frac{1}{12} x^{2}+6 \ln |x|+C121​x2+6ln∣x∣+C 
B)    16x+C\frac{1}{6} x+C61​x+C 
C)    x+Cx+Cx+C 
D)    xln⁡6+6ln⁡∣x∣+Cx \ln 6+6 \ln |x|+Cxln6+6ln∣x∣+CE) C) and D)
F) A) and B)

Question 3

Find the particular solution of the differential equation.
- dydx=4x;y=13\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{4}{\mathrm{x}} ; \mathrm{y}=13dxdy​=x4​;y=13  when  x=1\mathrm{x}=1x=1

Accepted Answer

A)    y=4ln⁡x+2y=4 \ln x+2y=4lnx+2 
B)    y=ln⁡x+13y=\ln x+13y=lnx+13 
C)    y=ln⁡x+11y=\ln x+11y=lnx+11 
D)    y=4ln⁡x+13y=4 \ln x+13y=4lnx+13E) A) and B)
F) A) and C)

Question 4

Solve the problem.
-The work  WWW  (in joules)  done by a force  FFF  (in newtons)  moving an object through a distance  xxx  (in meters)  is given by  W=∫Fdx\mathrm{W}=\int \mathrm{Fdx}W=∫Fdx . Find a formula for  W\mathrm{W}W , if  F=kx\mathrm{F}=\mathrm{kx}F=kx  and  k\mathrm{k}k  is a constant.

Accepted Answer

A)    W=kx2+CW=\frac{k x}{2}+CW=2kx​+C 
B)    W=kx22+C\mathrm{W}=\frac{k x^{2}}{2}+CW=2kx2​+C 
C)    W=kxx2+C\mathrm{W}=\mathrm{kx} \mathrm{x}^{2}+\mathrm{C}W=kxx2+C 
D)    W=k+C\mathrm{W}=\mathrm{k}+\mathrm{C}W=k+CE) All of the above
F) A) and B)

Question 5

Evaluate the definite integral.
- ∫04et(10+et) 3dt\int_{0}^{4} \frac{\mathrm{e}^{\mathrm{t}}}{\left(10+\mathrm{e}^{\mathrm{t}}\right) ^{3}} d t∫04​(10+et) 3et​dt

Accepted Answer

A)   0.0160
B)   -0.0024
C)   -0.0080
D)   0.0040 E) B) and D)
F) None of the above

Question 6

Find the integral.
- ∫7pe(5p2) dp\int 7 p e^{\left(5 p^{2}\right) } d p∫7pe(5p2) dp

Accepted Answer

A)    7e(5p2) +C7 e^{\left(5 p^{2}\right) }+C7e(5p2) +C 
B)    −7e(5p2) +C-7 e^{\left(5 p^{2}\right) }+C−7e(5p2) +C 
C)    710e(5p2) +C\frac{7}{10} e^{\left(5 p^{2}\right) }+C107​e(5p2) +C 
D)    −75e(5p2) +C-\frac{7}{5} e^{\left(5 p^{2}\right) }+C−57​e(5p2) +CE) All of the above
F) A) and B)

Question 7

Use integration by parts to find the integral.
- ∫ln⁡6xx3dx\int \frac{\ln 6 x}{x^{3}} d x∫x3ln6x​dx

Accepted Answer

A)    −12x−2ln⁡6x−14x−2+C-\frac{1}{2} x^{-2} \ln 6 x-\frac{1}{4} x^{-2}+C−21​x−2ln6x−41​x−2+C 
B)    ln⁡6x+12x−2+C\ln 6 x+\frac{1}{2} x^{-2}+Cln6x+21​x−2+C 
C)    −12x−2ln⁡6x−12x−1+C-\frac{1}{2} x^{-2} \ln 6 x-\frac{1}{2} x^{-1}+C−21​x−2ln6x−21​x−1+C 
D)    −12x−2ln⁡6x+14x−2+C-\frac{1}{2} x^{-2} \ln 6 x+\frac{1}{4} x^{-2}+C−21​x−2ln6x+41​x−2+CE) B) and D)
F) None of the above

Question 8

Find the area of the shaded region.
-

Accepted Answer

A)    293\frac{29}{3}329​ 
B)    383\frac{38}{3}338​ 
C)    223\frac{22}{3}322​ 
D)    163\frac{16}{3}316​E) None of the above
F) C) and D)

Question 9

Find the integral.
- ∫dx(5+ln⁡x) x\int \frac{d x}{(5+\ln x)  x}∫(5+lnx) xdx​

Accepted Answer

A)    ln⁡∣5+ln⁡x∣+C\ln |5+\ln x|+Cln∣5+lnx∣+C 
B)    ln⁡(5+ln⁡x) +C\ln (5+\ln x) +Cln(5+lnx) +C 
C)    5ln⁡∣5+ln⁡x∣+C5 \ln |5+\ln x|+C5ln∣5+lnx∣+C 
D)    15ln⁡∣5+ln⁡x∣+C\frac{1}{5} \ln |5+\ln x|+C51​ln∣5+lnx∣+CE) A) and C)
F) A) and D)

Question 10

Find the given indefinite integral.
- ∫ln⁡(7x+8) dx\int \ln (7 x+8)  d x∫ln(7x+8) dx

Accepted Answer

A)    1756x−ln⁡(7x+8) −(7x+8) ]+C\left.\frac{1}{7} 56 x-\ln (7 x+8) -(7 x+8) \right]+C71​56x−ln(7x+8) −(7x+8) ]+C 
B)    17[(7x+8) ln⁡x−(7x+8) ]+C\frac{1}{7}[(7 x+8)  \ln x-(7 x+8) ]+C71​[(7x+8) lnx−(7x+8) ]+C 
C)    17[(7x+8) ln⁡(7x+8) −(7x+8) ]+C\frac{1}{7}[(7 x+8)  \ln (7 x+8) -(7 x+8) ]+C71​[(7x+8) ln(7x+8) −(7x+8) ]+C 
D)    17[(7x+8) ln⁡(7x+8) −(7x+8) −56]+C\frac{1}{7}[(7 x+8)  \ln (7 x+8) -(7 x+8) -56]+C71​[(7x+8) ln(7x+8) −(7x+8) −56]+CE) A) and B)
F) B) and D)

Question 11

Find the area bounded by the given curves.
- y=5xy=\frac{5}{x}y=x5​  and  y=1+5x−x2;[1,5]y=1+5 x-x^{2} ;[1,5]y=1+5x−x2;[1,5]

Accepted Answer

A)   14.619
B)   2.569
C)   46.379
D)   9.730 E) A) and B)
F) C) and D)

Question 12

Find the particular solution of the differential equation.
- 2dydx−4xy=8x;y=82 \frac{d y}{d x}-4 x y=8 x ; y=82dxdy​−4xy=8x;y=8  when  x=0x=0x=0

Accepted Answer

A)    y=−2+10e−x2y=-2+10 e^{-x^{2}}y=−2+10e−x2 
B)    y=−1+9ex2y=-1+9 e^{x^{2}}y=−1+9ex2 
C)    y=2+8ex2y=2+8 e^{x^{2}}y=2+8ex2 
D)    y=−2+10ex2y=-2+10 e^{x^{2}}y=−2+10ex2E) B) and D)
F) B) and C)

Question 13

Evaluate the integral.
- ∫14x1/2dx\int_{1}^{4} x^{1 / 2} d x∫14​x1/2dx

Accepted Answer

A)   2.8
B)   4.67
C)   3.4
D)   5.67 E) A) and D)
F) B) and D)

Question 14

Use integration by parts to find the integral.
- ∫3xexdx\int 3 x e^{x} d x∫3xexdx

Accepted Answer

A)    3xex−3ex+C3 x e^{x}-3 e^{x}+C3xex−3ex+C 
B)    3ex−ex+C3 e^{x}-e^{x}+C3ex−ex+C 
C)    3ex−3xex+C3 e^{x}-3 x e^{x}+C3ex−3xex+C 
D)    xex−3ex+Cx e^{x}-3 e^{x}+Cxex−3ex+CE) B) and C)
F) None of the above

Question 15

Find the area bounded by the given curves.
- y=2x−x2,y=2x−4y=2 x-x^{2}, y=2 x-4y=2x−x2,y=2x−4

Accepted Answer

A)    343\frac{34}{3}334​ 
B)    373\frac{37}{3}337​ 
C)    323\frac{32}{3}332​ 
D)    313\frac{31}{3}331​E) A) and D)
F) A) and C)

Question 16

Find the integral.
- ∫3(2t+5) 3dt\int 3(2 t+5) ^{3} d t∫3(2t+5) 3dt

Accepted Answer

A)    14(2t+5) 4+C\frac{1}{4}(2 t+5) ^{4}+C41​(2t+5) 4+C 
B)    38(2t+5) 4+C\frac{3}{8}(2 t+5) ^{4}+C83​(2t+5) 4+C 
C)    12(2t+5) 4+C\frac{1}{2}(2 t+5) ^{4}+C21​(2t+5) 4+C 
D)    34(2t+5) 4+C\frac{3}{4}(2 t+5) ^{4}+C43​(2t+5) 4+CE) A) and B)
F) B) and C)

Question 17

Find the integral.
- ∫6x−7dx\int \sqrt{6 x-7} d x∫6x−7​dx

Accepted Answer

A)    12(6x−7) 3/2+C\frac{1}{2}(6 x-7) ^{3 / 2}+C21​(6x−7) 3/2+C 
B)    19(6x−7) 3/2+C\frac{1}{9}(6 x-7) ^{3 / 2}+C91​(6x−7) 3/2+C 
C)    13(6x−7) 3/2+C\frac{1}{3}(6 x-7) ^{3 / 2}+C31​(6x−7) 3/2+C 
D)    16(6x−7) 3/2+C\frac{1}{6}(6 x-7) ^{3 / 2}+C61​(6x−7) 3/2+CE) None of the above
F) All of the above

Question 18

Work the exercise.
-Find the total revenue function  R(x) R(x) R(x)   (in thousands of dollars)  if the marginal revenue (in thousands of dollars per unit)  at a production level of  xxx  units is  R′=49x+3ex(0≤x≤7) R^{\prime}=\frac{49 x+3}{e^{x}}(0 \leq x \leq 7) R′=ex49x+3​(0≤x≤7)  , and  R(0) =0R(0) =0R(0) =0 .

Accepted Answer

A)    R(x) =49xe−x−10e−x−52R(x) =49 x e^{-x}-10 e^{-x}-52R(x) =49xe−x−10e−x−52 
B)    R(x) =49xe−x+52e−x−49R(x) =49 x e^{-x}+52 e^{-x}-49R(x) =49xe−x+52e−x−49 
C)    R(x) =−49xe−x+52e−x+49R(x) =-49 x e^{-x}+52 e^{-x}+49R(x) =−49xe−x+52e−x+49 
D)    R(x) =−49xe−x−52e−x+52R(x) =-49 x e^{-x}-52 e^{-x}+52R(x) =−49xe−x−52e−x+52E) A) and C)
F) B) and C)

Question 19

Find the integral.
- ∫x45x5+3dx\int \frac{x^{4}}{5 x^{5}+3} d x∫5x5+3x4​dx

Accepted Answer

A)    15ln⁡∣5x5+3∣+C\frac{1}{5} \ln \left|5 x^{5}+3\right|+C51​lnc10,0,16.667,-5,20,-15 v-585 v0 v-585 c-2.667,-10,-9.667,-15,-21,-15c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v0 v585 h43z'/>​5x5+3c10,0,16.667,-5,20,-15 v-585 v0 v-585 c-2.667,-10,-9.667,-15,-21,-15c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v0 v585 h43z'/>​+C 
B)    25ln⁡(5x5+3) +C25 \ln \left(5 x^{5}+3\right) +C25ln(5x5+3) +C 
C)    125ln⁡∣5x5+3∣+C\frac{1}{25} \ln \left|5 x^{5}+3\right|+C251​lnc10,0,16.667,-5,20,-15 v-585 v0 v-585 c-2.667,-10,-9.667,-15,-21,-15c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v0 v585 h43z'/>​5x5+3c10,0,16.667,-5,20,-15 v-585 v0 v-585 c-2.667,-10,-9.667,-15,-21,-15c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v0 v585 h43z'/>​+C 
D)    25ln⁡∣5x5+3∣+C25 \ln \left|5 x^{5}+3\right|+C25lnc10,0,16.667,-5,20,-15 v-585 v0 v-585 c-2.667,-10,-9.667,-15,-21,-15c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v0 v585 h43z'/>​5x5+3c10,0,16.667,-5,20,-15 v-585 v0 v-585 c-2.667,-10,-9.667,-15,-21,-15c-10,0,-16.667,5,-20,15z M188 15 H145 v585 v0 v585 h43z'/>​+CE) A) and B)
F) B) and C)

Question 20

Find the integral.
- ∫x5+1xdx\int \frac{x^{5}+1}{x} d x∫xx5+1​dx

Accepted Answer

A)    13x4−ln⁡∣x∣+C\frac{1}{3} x^{4}-\ln |x|+C31​x4−ln∣x∣+C 
B)    15x5+ln⁡∣x∣+C\frac{1}{5} x^{5}+\ln |x|+C51​x5+ln∣x∣+C 
C)    15x5−ln⁡∣x∣+C\frac{1}{5} x^{5}-\ln |x|+C51​x5−ln∣x∣+C 
D)    13x4+ln⁡∣x∣+C\frac{1}{3} x^{4}+\ln |x|+C31​x4+ln∣x∣+CE) B) and C)
F) B) and D)

Exam 13: Integral Calculus

Evaluate the integral. - ∫14x1/2dx\int_{1}^{4} x^{1 / 2} d x∫14​x1/2dx

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Find the area of the shaded region. -

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Find the area bounded by the given curves. - y=5xy=\frac{5}{x}y=x5​ and y=1+5x−x2;[1,5]y=1+5 x-x^{2} ;[1,5]y=1+5x−x2;[1,5]

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Find the integral. - ∫x45x5+3dx\int \frac{x^{4}}{5 x^{5}+3} d x∫5x5+3x4​dx

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Find the area bounded by the given curves. - y=2x−x2,y=2x−4y=2 x-x^{2}, y=2 x-4y=2x−x2,y=2x−4

Correct AnswerverifiedShow Answer

Find the integral. - ∫3(2t+5) 3dt\int 3(2 t+5) ^{3} d t∫3(2t+5) 3dt

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Find the integral. - ∫7pe(5p2) dp\int 7 p e^{\left(5 p^{2}\right) } d p∫7pe(5p2) dp

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Use integration by parts to find the integral. - ∫ln⁡6xx3dx\int \frac{\ln 6 x}{x^{3}} d x∫x3ln6x​dx

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Find the integral. - ∫6x−7dx\int \sqrt{6 x-7} d x∫6x−7​dx

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Evaluate the definite integral. - ∫04et(10+et) 3dt\int_{0}^{4} \frac{\mathrm{e}^{\mathrm{t}}}{\left(10+\mathrm{e}^{\mathrm{t}}\right) ^{3}} d t∫04​(10+et) 3et​dt

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Use integration by parts to find the integral. - ∫3xexdx\int 3 x e^{x} d x∫3xexdx

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Find the given indefinite integral. - ∫ln⁡(7x+8) dx\int \ln (7 x+8) d x∫ln(7x+8) dx

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Solve the problem. -The work WWW (in joules) done by a force FFF (in newtons) moving an object through a distance xxx (in meters) is given by W=∫Fdx\mathrm{W}=\int \mathrm{Fdx}W=∫Fdx . Find a formula for W\mathrm{W}W , if F=kx\mathrm{F}=\mathrm{kx}F=kx and k\mathrm{k}k is a constant.

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Find the integral. - ∫dx(5+ln⁡x) x\int \frac{d x}{(5+\ln x) x}∫(5+lnx) xdx​

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Find the integral. - ∫x5+1xdx\int \frac{x^{5}+1}{x} d x∫xx5+1​dx

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Find the particular solution of the differential equation. - dydx=4x;y=13\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{4}{\mathrm{x}} ; \mathrm{y}=13dxdy​=x4​;y=13 when x=1\mathrm{x}=1x=1

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Find the integral. - ∫(x6+6x) dx\int\left(\frac{x}{6}+\frac{6}{x}\right) d x∫(6x​+x6​) dx

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Find the particular solution of the differential equation. - 2dydx−4xy=8x;y=82 \frac{d y}{d x}-4 x y=8 x ; y=82dxdy​−4xy=8x;y=8 when x=0x=0x=0

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

Correct Answer
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Find the area of the shaded region. - $Find the area of the shaded region. - A) \frac{29}{3} B) \frac{38}{3} C) \frac{22}{3} D) \frac{16}{3}$

Correct Answer
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Find the area bounded by the given curves. - $y=\frac{5}{x}$ and $y=1+5 x-x^{2} ;[1,5]$

Correct Answer
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Find the integral. - $\int \frac{x^{4}}{5 x^{5}+3} d x$

Correct Answer
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Find the area bounded by the given curves. - $y=2 x-x^{2}, y=2 x-4$

Correct Answer
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Find the integral. - $\int 3(2 t+5) ^{3} d t$

Correct Answer
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Find the integral. - $\int 7 p e^{\left(5 p^{2}\right) } d p$

Correct Answer
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Use integration by parts to find the integral. - $\int \frac{\ln 6 x}{x^{3}} d x$

Correct Answer
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Find the integral. - $\int \sqrt{6 x-7} d x$

Correct Answer
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Evaluate the definite integral. - $\int_{0}^{4} \frac{\mathrm{e}^{\mathrm{t}}}{\left(10+\mathrm{e}^{\mathrm{t}}\right) ^{3}} d t$

Correct Answer
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Use integration by parts to find the integral. - $\int 3 x e^{x} d x$

Correct Answer
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Find the given indefinite integral. - $\int \ln (7 x+8) d x$

Correct Answer
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Solve the problem. -The work $W$ (in joules) done by a force $F$ (in newtons) moving an object through a distance $x$ (in meters) is given by $\mathrm{W}=\int \mathrm{Fdx}$ . Find a formula for $\mathrm{W}$ , if $\mathrm{F}=\mathrm{kx}$ and $\mathrm{k}$ is a constant.

Correct Answer
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Find the integral. - $\int \frac{d x}{(5+\ln x) x}$

Correct Answer
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Correct Answer
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Find the integral. - $\int \frac{x^{5}+1}{x} d x$

Correct Answer
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Correct Answer
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Find the particular solution of the differential equation. - $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{4}{\mathrm{x}} ; \mathrm{y}=13$ when $\mathrm{x}=1$

Correct Answer
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Find the integral. - $\int\left(\frac{x}{6}+\frac{6}{x}\right) d x$

Correct Answer
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Find the particular solution of the differential equation. - $2 \frac{d y}{d x}-4 x y=8 x ; y=8$ when $x=0$

Correct Answer
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