Question 1

Evaluate the integral $\int_{a-2}^{a} \frac{d x}{(a-x)^{0.9}}$ to 3 decimal places, if it converges.If not, enter DNC.

Accepted Answer

The answer of Evaluate the integral $\int_{a-2}^{a} \frac{d x}{(a-x)^{0.9}}$ to...

Question 2

$\int u^{-1} \ln u d u=(\ln u)^{2}+C$ .

Accepted Answer

A) True 
 B)False

Question 3

Integrate  ∫cos⁡23xsin⁡3xdx\int \cos ^{2} 3 x \sin 3 x d x∫cos23xsin3xdx  .

Accepted Answer

A)    cos⁡33x9+C\frac{\cos ^{3} 3 x}{9}+C9cos33x​+C 
B)    −cos⁡33x9+C-\frac{\cos ^{3} 3 x}{9}+C−9cos33x​+C 
C)    cos⁡33x3+C\frac{\cos ^{3} 3 x}{3}+C3cos33x​+C 
D)    −cos⁡33x27+C-\frac{\cos ^{3} 3 x}{27}+C−27cos33x​+C

Question 4

Find  ∫3y(y2+2) 3dy\int 3 y\left(y^{2}+2\right) ^{3} d y∫3y(y2+2) 3dy

Accepted Answer

A)    38y(y2+2) 4+C\frac{3}{8 y}\left(y^{2}+2\right) ^{4}+C8y3​(y2+2) 4+C 
B)    3y28(y2+2) 4+C\frac{3 y^{2}}{8}\left(y^{2}+2\right) ^{4}+C83y2​(y2+2) 4+C 
C)    38(y2+2) 4+C\frac{3}{8}\left(y^{2}+2\right) ^{4}+C83​(y2+2) 4+C 
D)    34(y2+2) 4+C\frac{3}{4}\left(y^{2}+2\right) ^{4}+C43​(y2+2) 4+C

Question 5

$\int \frac{7 x^{2}+6}{x^{2}+1} d x=7 x-\arctan x+C$ .

Accepted Answer

A) True 
 B)False

Question 6

If $\int_{0}^{\infty} x^{2} e^{-x} d x$ converges, find its value.Otherwise, enter "DNC".

Accepted Answer

The answer of If $\int_{0}^{\infty} x^{2} e^{-x} d x$ converges,...

Question 7

Show $\int \frac{1}{x^{2}+a^{2}} d x=\frac{1}{a} \arctan \frac{x}{a}+C$ by integrating the left hand side using the substitution $x=a \tan \theta$ .

Accepted Answer

If <img  loading="lazy" src="/assets/images/blured-textbook.svg" class="d-inline m-1" alt="blured image" width="139" height="30"> , then...

Question 8

Evaluate exactly: $\int_{0}^{ x} x^{2} \sin x d x$ .(Leave $\pi$ in your answer.)

Accepted Answer

The answer of Evaluate exactly: \(\int_{0}^{ x} x^{2} \sin x...

Question 9

If $\int_{0}^{\infty} \frac{8}{\sqrt{1+x}} d x$ converges, find its value to 3 decimal places.Otherwise, enter "DNC".

Accepted Answer

The answer of If $\int_{0}^{\infty} \frac{8}{\sqrt{1+x}} d x$ converges, find...

Question 10

The table below shows the velocity v(t) of a falling object at various times (time t measured in seconds, velocity v(t) measured in meters per second) .  t012v(t) 172328\begin{array}{cccc}t & 0 & 1 & 2 \v(t)  & 17 & 23 & 28\end{array}tv(t) ​017​123​228​  The distance the object fell in these three seconds lies within which interval?

Accepted Answer

A)  (68, 75.5) 
B)  (75.5, 78) 
C)  (78, 83) 
D)  (83, 86.5)

Question 11

For any given function, TRAP(n)is always more accurate than LEFT(n).

Accepted Answer

A) True 
 B)False

Question 12

$\int_{-2}^{-1} \sin x^{2} d x>3$ .

Accepted Answer

A) True 
 B)False

Question 13

Is the area between $y=\frac{1}{x^{2}}$ and $y=\frac{1}{x^{5}}$ on (1, $\infty$)finite or infinite?

Accepted Answer

The answer of Is the area between $y=\frac{1}{x^{2}}$ and $y=\frac{1}{x^{5}}$...

Question 14

$\int\left(\sin ^{3} 2 \theta+5\right) \cos 2 \theta d \theta=\frac{1}{4} \sin ^{4} 2 \theta+\frac{5}{2} \sin 2 \theta+C$ .

Accepted Answer

A) True 
 B)False

Question 15

Consider the definite integral  ∫02/514+25x2dx\int_{0}^{2 / 5} \frac{1}{4+25 x^{2}} d x∫02/5​4+25x21​dx  .Compute the integral using the fundamental theorem of calculus and using the midpoint rule with n = 20.How far apart are your answers?

Accepted Answer

A)  Within 0.03 but not within 0.003
B)  Within 0.003 but not within 0.0003
C)  Within 0.0003 but not within 0.00003
D)  Within 0.00003 but not within 0.000003

Question 16

$\int(x-1) e^{-x} d x=-x e^{-x}-e^{-x}+C$ .

Accepted Answer

A) True 
 B)False

Question 17

Use the table of antiderivatives to determine if the following statement is true. $\int e^{7 x} \cos 8 x d x=\frac{1}{113} e^{7 x}(7 \cos 8 x+8 \sin 8 x)+C$

Accepted Answer

A) True 
 B)False

Question 18

Find  ∫5x4−x2dx\int \frac{5 x}{\sqrt{4-x^{2}}} d x∫4−x2​5x​dx

Accepted Answer

A)    −103(4−x2) 3/2+C-\frac{10}{3\left(4-x^{2}\right) ^{3 / 2}}+C−3(4−x2) 3/210​+C 
B)    103(4−x2) 3/2+C\frac{10}{3\left(4-x^{2}\right) ^{3 / 2}}+C3(4−x2) 3/210​+C 
C)    54−x2+C5 \sqrt{4-x^{2}}+C54−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)    −54−x2+C-5 \sqrt{4-x^{2}}+C−54−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 19

$\int \frac{7+x}{\sqrt{x}} d x=14 \sqrt{x}+\frac{2}{3} x^{\frac{3}{2}}+C$ .

Accepted Answer

A) True 
 B)False

Question 20

$\int_{0}^{3} \sin ^{39}(x) d x>\pi$ .

Accepted Answer

A) True 
 B)False

Exam 7: Integration

If ∫0∞x2e−xdx\int_{0}^{\infty} x^{2} e^{-x} d x∫0∞​x2e−xdx converges, find its value.Otherwise, enter "DNC".

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∫−2−1sin⁡x2dx>3\int_{-2}^{-1} \sin x^{2} d x>3∫−2−1​sinx2dx>3 .

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∫7+xxdx=14x+23x32+C\int \frac{7+x}{\sqrt{x}} d x=14 \sqrt{x}+\frac{2}{3} x^{\frac{3}{2}}+C∫x​7+x​dx=14x​+32​x23​+C .

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For any given function, TRAP(n)is always more accurate than LEFT(n).

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Evaluate the integral ∫a−2adx(a−x)0.9\int_{a-2}^{a} \frac{d x}{(a-x)^{0.9}}∫a−2a​(a−x)0.9dx​ to 3 decimal places, if it converges.If not, enter DNC.

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Integrate ∫cos⁡23xsin⁡3xdx\int \cos ^{2} 3 x \sin 3 x d x∫cos23xsin3xdx .

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∫7x2+6x2+1dx=7x−arctan⁡x+C\int \frac{7 x^{2}+6}{x^{2}+1} d x=7 x-\arctan x+C∫x2+17x2+6​dx=7x−arctanx+C .

Correct AnswerverifiedShow Answer

Correct AnswerverifiedShow Answer

∫(x−1)e−xdx=−xe−x−e−x+C\int(x-1) e^{-x} d x=-x e^{-x}-e^{-x}+C∫(x−1)e−xdx=−xe−x−e−x+C .

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Find ∫3y(y2+2) 3dy\int 3 y\left(y^{2}+2\right) ^{3} d y∫3y(y2+2) 3dy

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If ∫0∞81+xdx\int_{0}^{\infty} \frac{8}{\sqrt{1+x}} d x∫0∞​1+x​8​dx converges, find its value to 3 decimal places.Otherwise, enter "DNC".

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Evaluate exactly: ∫0xx2sin⁡xdx\int_{0}^{ x} x^{2} \sin x d x∫0x​x2sinxdx .(Leave π\piπ in your answer.)

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Is the area between y=1x2y=\frac{1}{x^{2}}y=x21​ and y=1x5y=\frac{1}{x^{5}}y=x51​ on (1, ∞\infty∞ )finite or infinite?

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Use the table of antiderivatives to determine if the following statement is true. ∫e7xcos⁡8xdx=1113e7x(7cos⁡8x+8sin⁡8x)+C\int e^{7 x} \cos 8 x d x=\frac{1}{113} e^{7 x}(7 \cos 8 x+8 \sin 8 x)+C∫e7xcos8xdx=1131​e7x(7cos8x+8sin8x)+C

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Show ∫1x2+a2dx=1aarctan⁡xa+C\int \frac{1}{x^{2}+a^{2}} d x=\frac{1}{a} \arctan \frac{x}{a}+C∫x2+a21​dx=a1​arctanax​+C by integrating the left hand side using the substitution x=atan⁡θx=a \tan \thetax=atanθ .

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∫u−1ln⁡udu=(ln⁡u)2+C\int u^{-1} \ln u d u=(\ln u)^{2}+C∫u−1lnudu=(lnu)2+C .

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∫(sin⁡32θ+5)cos⁡2θdθ=14sin⁡42θ+52sin⁡2θ+C\int\left(\sin ^{3} 2 \theta+5\right) \cos 2 \theta d \theta=\frac{1}{4} \sin ^{4} 2 \theta+\frac{5}{2} \sin 2 \theta+C∫(sin32θ+5)cos2θdθ=41​sin42θ+25​sin2θ+C .

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Consider the definite integral ∫02/514+25x2dx\int_{0}^{2 / 5} \frac{1}{4+25 x^{2}} d x∫02/5​4+25x21​dx .Compute the integral using the fundamental theorem of calculus and using the midpoint rule with n = 20.How far apart are your answers?

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Find ∫5x4−x2dx\int \frac{5 x}{\sqrt{4-x^{2}}} d x∫4−x2​5x​dx

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∫03sin⁡39(x)dx>π\int_{0}^{3} \sin ^{39}(x) d x>\pi∫03​sin39(x)dx>π .

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If $\int_{0}^{\infty} x^{2} e^{-x} d x$ converges, find its value.Otherwise, enter "DNC".

Correct Answer
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$\int_{-2}^{-1} \sin x^{2} d x>3$ .

Correct Answer
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$\int \frac{7+x}{\sqrt{x}} d x=14 \sqrt{x}+\frac{2}{3} x^{\frac{3}{2}}+C$ .

Correct Answer
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Correct Answer
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Evaluate the integral $\int_{a-2}^{a} \frac{d x}{(a-x)^{0.9}}$ to 3 decimal places, if it converges.If not, enter DNC.

Correct Answer
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Integrate $\int \cos ^{2} 3 x \sin 3 x d x$ .

Correct Answer
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$\int \frac{7 x^{2}+6}{x^{2}+1} d x=7 x-\arctan x+C$ .

Correct Answer
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Correct Answer
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$\int(x-1) e^{-x} d x=-x e^{-x}-e^{-x}+C$ .

Correct Answer
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Find $\int 3 y\left(y^{2}+2\right) ^{3} d y$

Correct Answer
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If $\int_{0}^{\infty} \frac{8}{\sqrt{1+x}} d x$ converges, find its value to 3 decimal places.Otherwise, enter "DNC".

Correct Answer
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Evaluate exactly: $\int_{0}^{ x} x^{2} \sin x d x$ .(Leave $\pi$ in your answer.)

Correct Answer
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Is the area between $y=\frac{1}{x^{2}}$ and $y=\frac{1}{x^{5}}$ on (1, $\infty$ )finite or infinite?

Correct Answer
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Use the table of antiderivatives to determine if the following statement is true. $\int e^{7 x} \cos 8 x d x=\frac{1}{113} e^{7 x}(7 \cos 8 x+8 \sin 8 x)+C$

Correct Answer
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Show $\int \frac{1}{x^{2}+a^{2}} d x=\frac{1}{a} \arctan \frac{x}{a}+C$ by integrating the left hand side using the substitution $x=a \tan \theta$ .

Correct Answer
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If , then...
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$\int u^{-1} \ln u d u=(\ln u)^{2}+C$ .

Correct Answer
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$\int\left(\sin ^{3} 2 \theta+5\right) \cos 2 \theta d \theta=\frac{1}{4} \sin ^{4} 2 \theta+\frac{5}{2} \sin 2 \theta+C$ .

Correct Answer
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Consider the definite integral $\int_{0}^{2 / 5} \frac{1}{4+25 x^{2}} d x$ .Compute the integral using the fundamental theorem of calculus and using the midpoint rule with n = 20.How far apart are your answers?

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Find $\int \frac{5 x}{\sqrt{4-x^{2}}} d x$

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$\int_{0}^{3} \sin ^{39}(x) d x>\pi$ .

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