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MathQuiz 4.3
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Quiz 4: Integration by Parts

Question

Question 1

Recall that integration by parts is a technique to re-express the integral of a product of two functions $u$ and $ddvx$ in a form which allows it to be more easily evaluated. The formula is $∫ udvdx = uv - ∫ vdudx dx dx$ . When applying the method of integration by parts to find $∫ xex dx$ , the best choice of $u$ and $dv dx$ is

a)	$u = ex, dv = x dx$	b)	$u = xex, dv= 1 dx$
c)	$u = 1, dv-= xex dx$	d)	$u = x, dv-= ex dx$
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 $∫ xex dx = 1x2ex - ∫ 1x2exdx 2 2$ .

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 $∫ xex dx = x2ex - ∫ (x2ex + xex)dx$ .

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 $∫ 1xex dx 0$ using integration by parts, and enter your answer.

Your answer is correct

Choosing $u = x, dv= ex dx$ gives $∫ 1xex dx = [xex]1- ∫1exdx = [ex(x- 1)]1= 1. 0 0 0 0$

Not correct. You may try again.

Try choosing $dv x u = x, dx = e$ .

Question 3

The reduction formula for $In = ∫ x(ln x)ndx$ is $In = 1x2(lnx)n - nIn-1 2 2$ . Given that $I = 1x2 + C 0 2$ , find $∫ ex(ln x)dx 1$ . Give your answer correct to three decimal places.

Your answer is correct

The integral $I1$ is $1 2 1 2 2x lnx - 4x + C$ , 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 $I = ∫ xnexdx = xnex - nI n n-1$ 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 $∫1x tan- 1xdx 0$ ? (Hint: use integration by parts with $-1 u = tan x$ and $dv dx = x$ .)

a)	$π- 1 4 - 2$	b)	$1 -2$
c)	$π- 1 8 - 2$	d)	$1 π- 2 (4 + 1)$
e)	$π- 4$

Your answer is correct.

The indefinite integral is $12x2 tan- 1x- 12x + 12 tan-1 x+ C$ , 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 $∫ 12 sin-1 xdx 0$ ? (Hint: use integration by parts with $u = sin-1 x$ and $ddvx = 1$ .)

a)	$√- -π - -3-+ 1 12 2$	b)	$√- π-+ -3- 4 2$
c)	$√- -π + -3-- 1 12 2$	d)	$π-- 1 3$
e)	$√ - -π - --3- 1 12 2$

Not correct. Choice (a) is false.

Not correct. Choice (b) is false.

Your answer is correct.

The indefinite integral is $-1 √-----2 x sin x + 1 - x + C$ , which, when evaluated between $0$ and $12$ , matches this option.

Not correct. Choice (d) is false.

Not correct. Choice (e) is false.

Question 7

The finite area bounded by the curve $y = ln x$ , the line $y = 1$ and the tangent line to $y = ln x$ at $x = 1$ is given as an integral with respect to $x$ by $∫ ∫ 12(x - 1- lnx)dx+ 2e(1 - ln x)dx$ . 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)	$∫ 01lny - y + 1dy$
c)	$∫ 0eey - y+ 1dy$	d)	$∫ 1ey - (y+ 1)dy 0$
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 $∫ x sec2 xdx$ ?

a)	$xsecxtan x- ln (cosx)+ C$	b)	$x tan x+ ln\|cosx\|+ C$
c)	$xtan2x - ln\|cosx\|+ C$	d)	$xtan x- ln (cosx)+ C$
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)	$∫ x sin(lnx)- cos(lnx )dx$	b)	$- cosxlnx + ∫ cosx dx x$
c)	$∫ - x sin(lnx)+ xcos(ln x)dx$	d)	$∫ xsin(ln x)- sin(lnx)dx x$
e)	$∫ cos x cosx lnx + -x--dx$