Quiz 4: Integration by Parts

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Question 1

Recall that integration by parts is a technique to re-express the integral of a product of two functions u and dv dx in a form which allows it to be more easily evaluated. The formula is udv dxdx = uv -vdu dxdx. When applying the method of integration by parts to find xexdx, the best choice of u and dv dx is
a)
u = ex,dv dx = x
  b)
u = xex,dv dx = 1
c)
u = 1,dv dx = xex
  d)
u = x,dv dx = ex
e)
None of the above

 

Not correct. Choice (a) is false.
This expresses the original integral in terms of a new integral which is even harder! After applying integration by parts as suggested, we have xexdx = 1 2x2ex -1 2x2exdx.
Not correct. Choice (b) is false.
This expresses the original integral in terms of a new integral which is even harder! After applying integration by parts as suggested, we have xexdx = x2ex -(x2ex + xex)dx.
Not correct. Choice (c) is false.
The problem of finding v if dv = xexdx is exactly the integral we started with! So no progress in this case.
Your answer is correct.
This gives genuine simplification and results in an easy integral. We obtain xexdx = xex -exdx.
Not correct. Choice (e) is false.

Question 2

Find 01xexdx using integration by parts, and enter your answer.

 

Your answer is correct
Choosing u = x,dv dx = ex gives 01xexdx = [xex]01 -01exdx = [ex(x - 1)]01 = 1.
Not correct. You may try again.
Try choosing u = x,dv dx = ex.

Question 3

The reduction formula for In =x(lnx)ndx is In = 1 2x2(lnx)n -n 2 In-1. Given that I0 = 1 2x2 + C, find 1ex(lnx)dx. Give your answer correct to three decimal places.

 

Your answer is correct
The integral I1 is 1 2x2 lnx -1 4x2 + C, using the reduction formula with n = 1. Evaluating this between 1 and e gives 2.097 to three decimal places.
Not correct. You may try again.
Substitute n = 1 into the reduction formula to obtain I1.

Question 4

Which option is an antiderivative of x4ex ? Use the reduction formula In =xnexdx = xnex - nIn-1 to help answer this question.
a)
exx4 + 4x3ex - 12x2ex + 24xex - 24ex
  b)
ex(x4 - 4x3 - 12x2 - 24x + 24)
c)
exx4 - 4x3ex + 12x2ex - 24xex + 24ex + Cex
  d)
ex(x4 - 4x3 + 12x2 - 24x + 24) + 15
e)
None of the above

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
Not correct. Choice (e) is false.

Question 5

Which option equals 01xtan-1xdx ? (Hint: use integration by parts with u = tan-1x and dv dx = x.)
a)
π 4 - 1 2
  b)
-1 2
c)
π 8 - 1 2
  d)
1 2(π 4 + 1)
e)
π 4

 

Your answer is correct.
The indefinite integral is 1 2x2 tan-1x -1 2x + 1 2 tan-1x + C, which, when evaluated between 0 and 1, matches this option.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.

Question 6

Which option equals 01 2 sin-1xdx ? (Hint: use integration by parts with u = sin-1x and dv dx = 1.)
a)
π 12 -3 2 + 1
  b)
π 4 + 3 2
c)
π 12 + 3 2 - 1
  d)
π 3 - 1
e)
π 12 -3 2 - 1

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
The indefinite integral is xsin-1x + 1 - x2 + C, which, when evaluated between 0 and 1 2, matches this option.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.

Question 7

The finite area bounded by the curve y = lnx, the line y = 1 and the tangent line to y = lnx at x = 1 is given as an integral with respect to x by 12(x - 1 - lnx)dx +2e(1 - lnx)dx. Which option equals the same area given as an integral with respect to y ? (You must draw a sketch to help you with this question.)
a)
1eey - (y + 1)dy
  b)
01 lny - y + 1dy
c)
0eey - y + 1dy
  d)
01ey - (y + 1)dy
e)
None of the above

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Not correct. Choice (c) is false.
Your answer is correct.
Not correct. Choice (e) is false.

Question 8

Which option equals xsec2xdx ?
a)
xsecxtanx - ln(cosx) + C
  b)
xtanx + ln|cosx| + C
c)
xtan2x - ln|cosx| + C
  d)
xtanx - ln(cosx) + C
e)
None of the above

 

Not correct. Choice (a) is false.
Your answer is correct.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.

Question 9

In some problems you need to apply the integration by parts method twice in order to obtain the required answer. The integral sin(lnx)dx is one such problem. Which of the following options gives the expression obtained after one application of integration by parts?
a)
xsin(lnx) -cos(lnx)dx
  b)
-cosxlnx +cosx x dx
c)
-xsin(lnx) +xcos(lnx)dx
  d)
xsin(lnx) -sin(lnx) x dx
e)
cosxlnx +cosx x dx

 

Your answer is correct.
Not correct. Choice (b) is false.
Try u = sin(lnx) and dv dx = 1 in the integration by parts formula.
Not correct. Choice (c) is false.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.

Question 10

Which option equals 1e sin(lnx)dx  ?
a)
1 2(esin1 + ecos1)
  b)
sin1 - cos1
c)
1 2(esin1 - ecos1 + 1)
  d)
1 2sin1 - 2ecos1 + 1
e)
esin1 - ecos1

 

Not correct. Choice (a) is false.
Not correct. Choice (b) is false.
Your answer is correct.
Not correct. Choice (d) is false.
Not correct. Choice (e) is false.
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