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### 14 Techniques of Integration

Course: MATH MAT1300, Spring 2010
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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 . __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ VET:Calculus 81 14.TechniquesofIntegration __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ VET:Calculus 82 14.TechniquesofIntegration 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 . VET:Calculus 83 14.TechniquesofIntegration __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ VET:Calculus 84 14.TechniquesofIntegration 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. __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ VET:Calculus 85 14.TechniquesofIntegration __________________________________________________________________________________ __________________________________________________________________________________ VET:Calculus 86 14.TechniquesofIntegration 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 __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ VET:Calculus 87 14.TechniquesofIntegration 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 __________________________________________________________________________________ __________________________________________________________________________________ VET:Calculus 88 14.TechniquesofIntegration __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ VET:Calculus 89 14.TechniquesofIntegration 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. __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ VET:Calculus 90 14.TechniquesofIntegration (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? __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ VET:Calculus 91 14.TechniquesofIntegration VET:Calculus 92 14.TechniquesofIntegration 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 __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ __________________________________________________________________________________ VET:Calculus 93 14.TechniquesofIntegration __________________________________________________________________________________ __________________________________________________________________________________ VET:Calculus 94 14.TechniquesofIntegration 9. Determinethevalueofasothat 1 a dx =10. x [3] _________________________________________________________________________________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________________________________________________________________________________ 10. Evaluatethefollowingintegrals. (a) e2 x 1 e 4x dx . (Hint:First,letu=e2xandthenletsin=u.) [5] _________________________________________________________________________________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________________________________________________________________________________ VET:Calculus 95 14.TechniquesofIntegration _________________________________________________________________________________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________________________________________________________________________________ VET:Calculus 96 14.TechniquesofIntegration (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] _________________________________________________________________________________________________________________________________________________________________________________________________________________________ VET:Calculus 97 14.TechniquesofIntegration _________________________________________________________________________________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________________________________________________________________________________ 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] _________________________________________________________________________________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________________________________________________________________________________ _________________________________________________________________________________________________________________________________________________________________________________________________________________________ VET:Calculus 98 14.TechniquesofIntegration
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