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18 Pages
14 Techniques of Integration
Course: MATH MAT1300, Spring 2010School: Acadia
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Word Count: 505
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TECHNIQUESOFINTEGRATION 1. 14. Evaluatethefollowingintegrals. (a) [5marks] /6 0 sin 3 2 x cos 2 x dx . __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________...
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TECHNIQUESOFINTEGRATION
1. 14. Evaluatethefollowingintegrals. (a) [5marks]
/6
0
sin 3 2 x cos 2 x dx .
__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________
(b) [5marks]
4
0
x 1 dx .
__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________
(c) [3marks]
/4
0
e 2i d .
__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________
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__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________
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2. Evaluatethefollowingintegrals.
/6
(a) [4marks]
cos x e
0
2 sin x
dx
__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________
3
(b) [3marks]
1
x 10 x 2
dx
__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________
1
(c) [3marks]
dx
0
d
x 4 + 1 dx .
__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ 3. (a) [2marks] Differentiatey=xlnxwithrespecttox.
__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ (b) [3marks] Useyouranswertofind ln x dx .
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__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________
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4. Findthefollowingintegrals. (a) [4marks]
x 3 + x 2 sin (x) dx x2
__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________
(b) [4marks]
x
2
4 x + 12 dx + 6x + 9
__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ x 3 e x ( 3 x 2 + 1 ) dx
(c) [4marks]
e
__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________
(d) [4marks]
6e
0
4i d
givingyouranswerincartesianform.
__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________
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__________________________________________________________________________________ __________________________________________________________________________________
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5. (a) [2marks] Statethedomainof f ( x) =
(
x +1 x 1
)
2
__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ (b) [2marks] Expandandsimplifytheexpression
(
x +1 x 1
)
2
__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________
(c) [3marks]
Useyourresultfrom(b)toevaluate
2 x 2 x 2 1 dx
__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________
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6. Findthefollowingintegrals. (a) [4marks]
(e
4x
+
1 1) dx 2x 3
__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________
(b) [3marks]
cos(ln( x )) dx x
__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________
(c) [4marks]
tan x(1 + cos
2
x) dx
__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________
/6
(d) [3marks]
0
1 2 sin 2 x dx
__________________________________________________________________________________ __________________________________________________________________________________
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__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________
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7. Thefunctionfisdefinedby f (t ) = sin(tx)dx
0 2
(a) [4marks]
1 cos(2t ) for t 0 Showthat f (t ) = t for t = 0 0
__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________
(b) [3marks]
Determine
lim f (t )
t 0
,justifyingyouranswer.
__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________
(c) [2marks]
Isfcontinuousatt=0?Justifyyouranswer.
__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________
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(d) [4marks]
Sketchthegraphoff.
f(t) 2 1 t
-10 -8 -6 -4 -2 -1 -2
2 4 6 8 10
(e) [1mark] WhatistheleastintegervalueofKsuchthatallsolutionsoftheequation f(t)=0.25arecontainedintheinterval[0,K]? __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ (f) [2marks] Howmanyvaluesoftaretherewithf(t)=0.25?
__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________
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8.
(a) 3t [4marks]
t + t 3t dt
__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________
2x (b) [4marks] 0 (e 6 sin 2 x) dx
P
__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________
(a) [4marks]
x dx (1 + x 2 ) 2
__________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________
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__________________________________________________________________________________ __________________________________________________________________________________
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9. Determinethevalueofasothat
1
a
dx =10. x
[3]
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
10. Evaluatethefollowingintegrals. (a)
e2 x 1 e
4x
dx . (Hint:First,letu=e2xandthenletsin=u.)
[5]
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
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_________________________________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
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(b)
2 2 sin x dx. x + cos x
(Hint:Letu=x+cosx.)
[3]
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
11.
(a) Showthat
x
0
1
n
sin( 2x )dx >0foralln 0.
[3]
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
(b) Alsoshowthat x
0
1
n
sin( 2x )dx <
x dx = n + 1 .
n 0
1
1
[4]
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
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_________________________________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
n (c) Useagraphicscalculatortographthefunctionf(x)= x sin( x ) for0 x 1andforn=0,1,2,3,4. 2
n x Hence,obtainapproximationsfor x sin( 2 )dx . 0
1
Recordyourresultsinthetablebelow.
[4]
n 0 1 2 3 4
n x sin( 2x )dx 0
1
1 n +1
(d) Basedonyourfindingsinpart(c),determinethevaluesofpandqwhere
999 x p< x sin( 2 )dx <q. 0
1
[2]
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
_________________________________________________________________________________________________________________________________________________________________________________________________________________________
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