Integration by substitution - Mathcentre

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ExampleSupposewewishtofind∫11 − 2x dx.Wemakethesubstitution u = 1 − 2xinordertosimplifytheintegrandto 1. Recallthattheuintegralof 1 withrespectto uisthenaturallogarithmof u, ln |u|.Asbefore,udu =( ) dudxdxandsoItfollowsthatwith u = 1 − 2xdu =and( ) dudx = −2 dxdxdudx = −2Theintegralbecomes∫ 1u(− 1 2 du )= − 1 2∫ 1u du= − 1 ln |u| + c2= − 1 ln |1 − 2x| + c2Theresultofthepreviousexamplecanbegeneralised: ifwewanttofindsubstitution u = ax + bleadsto 1 ∫ 1a u duwhichequals 1 ln |ax + b| + c.a∫1ax + b dx,theThismeans,forexample,thatwhenfacedwithanintegralsuchas13x + 7 dxwecanimmediatelywritedowntheansweras1 ln |3x + 7| + c.3∫∫Key Point1ax + b dx = 1 ln |ax + b| + cawww.mathcentre.ac.uk 4 c○mathcentre2009
Alittlemorecaremustbetakenwiththelimitsofintegrationwhendealingwithdefiniteintegrals.Considerthefollowingexample.ExampleSupposewewishtofind∫ 31(9 + x) 2 dxWemakethesubstitution u = 9 + x.Asbefore,( ) dudu = dxdxandsoItfollowsthatTheintegralbecomeswith u = 9 + xdu =and( ) dudx = dxdx∫ x=3x=1u 2 dududx = 1wherewehaveexplicitlywrittenthevariableinthelimitsofintegrationtoemphasisethatthoselimitswereonthevariable xandnot u.Wecanwritetheseaslimitson uusingthesubstitutionu = 9 + x.Clearly,when x = 1, u = 10,andwhen x = 3, u = 12.Sowerequire∫ u=12u=10u 2 du =[ 13 u3 ] 1210= 1 (12 3 − 10 3)3= 7283Notethatinthisexamplethereisnoneedtoconverttheanswergivenintermsof ubackintooneintermsof xbecausewehadalreadyconvertedthelimitson xintolimitson u.Exercises1.1.Ineachcaseuseasubstitutiontofindtheintegral:∫∫ 1∫(a) (x − 2) 3 dx (b) (x + 5) 4 dx (c) (2x − 1) 7 dx (d)02.Ineachcaseuseasubstitutiontofindtheintegral:∫∫ ∫ π/2(a) sin(7x − 3)dx (b) e 3x−2 dx (c) cos(1 − x)dx0∫ 1−1∫(d)(1 − x) 3 dx.17x + 5 dx.www.mathcentre.ac.uk 5 c○mathcentre2009
Page 1 and 2: Integrationby substitutionmc-TY-int
Page 3: ExampleSupposenowwewishtofindtheint
Page 7 and 8: Further,wenotethatif g(x) = 1 + x 2
Page 9 and 10: Writing f(u) = sin uand g(x) = √

Integration by substitution - Mathcentre

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