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MATH1003 Quizzes

Quiz 4: Integration by Parts
Question 1 Questions
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 Exactly one option must be correct)
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

Choice (a) is incorrect
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.
Choice (b) is incorrect
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.
Choice (c) is incorrect
The problem of finding v if dv = xexdx is exactly the integral we started with! So no progress in this case.
Choice (d) is correct!
This gives genuine simplification and results in an easy integral. We obtain xexdx = xex exdx.
Choice (e) is incorrect
Find 01xexdx using integration by parts, and enter your answer.

Correct!
Choosing u = x,dv dx = ex gives 01xexdx = [xex]01 01exdx = [ex(x 1)]01 = 1.
Incorrect. Please try again.
Try choosing u = x,dv dx = ex.
The reduction formula for In =x(lnx)ndx is In = 1 2x2(lnx)n n 2 In1. Given that I0 = 1 2x2 + C, find 1ex(lnx)dx. Give your answer correct to three decimal places.

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.
Incorrect. Please try again.
Substitute n = 1 into the reduction formula to obtain I1.
Which option is an antiderivative of x4ex ? Use the reduction formula In =xnexdx = xnex nIn1 to help answer this question. Exactly one option must be correct)
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

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Choice (e) is incorrect
Which option equals 01xtan1xdx ? (Hint: use integration by parts with u = tan1x and dv dx = x.) Exactly one option must be correct)
a)
π 4 1 2
b)
1 2
c)
π 8 1 2
d)
1 2(π 4 + 1)
e)
π 4

Choice (a) is correct!
The indefinite integral is 1 2x2 tan1x 1 2x + 1 2 tan1x + C, which, when evaluated between 0 and 1, matches this option.
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
Which option equals 01 2 sin1xdx ? (Hint: use integration by parts with u = sin1x and dv dx = 1.) Exactly one option must be correct)
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

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
The indefinite integral is xsin1x + 1 x2 + C, which, when evaluated between 0 and 1 2, matches this option.
Choice (d) is incorrect
Choice (e) is incorrect
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.) Exactly one option must be correct)
a)
1eey (y + 1)dy
b)
01 lny y + 1dy
c)
0eey y + 1dy
d)
01ey (y + 1)dy
e)
None of the above

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is incorrect
Choice (d) is correct!
Choice (e) is incorrect
Which option equals xsec2xdx ? Exactly one option must be correct)
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

Choice (a) is incorrect
Choice (b) is correct!
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
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? Exactly one option must be correct)
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

Choice (a) is correct!
Choice (b) is incorrect
Try u = sin(lnx) and dv dx = 1 in the integration by parts formula.
Choice (c) is incorrect
Choice (d) is incorrect
Choice (e) is incorrect
Which option equals 1e sin(lnx)dx  ? Exactly one option must be correct)
a)
1 2(esin1 + ecos1)
b)
sin1 cos1
c)
1 2(esin1 ecos1 + 1)
d)
1 2sin1 2ecos1 + 1
e)
esin1 ecos1

Choice (a) is incorrect
Choice (b) is incorrect
Choice (c) is correct!
Choice (d) is incorrect
Choice (e) is incorrect