Integration that leads to logarithm functions - Mathcentre
Integration that leads to logarithm functions - Mathcentre
Integration that leads to logarithm functions - Mathcentre
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
Example∫ x cosx + sin xFinddx.x sin xFirs<strong>to</strong>fallthinkaboutwhatwewouldobtainifwedifferentiatedthedenomina<strong>to</strong>r:let’sdothisfirst.If y = x sin x,thenusingtheproductruleofdifferentiation,dydx= x cosx + sin x.Sowesee<strong>that</strong>intheintegralwearetrying<strong>to</strong>find, thenumera<strong>to</strong>risthederivativeofthedenomina<strong>to</strong>r.So∫ x cosx + sin xdx = ln |x sin x| + c.x sin xExercises1.Determineeachofthefollowingintegrals∫∫∫3a)2 + 3x dx b) x1 + 2x dx c) e 2x2 e 2x + 1 dx∫e 2x ∫∫d)e 2x − 1 dx e) x −3cotxdx f)x −2 + 4 dx∫ f′2.Foreachofthefollowingintegrals,usetheresult = ln |f|+c<strong>to</strong>determinetheintegral.fThenrepeattheintegral,usingalgebra<strong>to</strong>simplifytheintegrandbeforeintegration.Check<strong>that</strong>thetwoanswersobtainedarethesame.∫ 3x2∫ cosxesinx∫ 4x−5a)x dx b) dx c) dx3 e sin x x−4 ∫ x−1/2∫ 5e5x∫ x−5/2d) dx2x e) 1/2e dx f) dx5x x−3/2 Answers1.1a) ln |2 + 3x| + c b)4 ln ∣ ∣ 1 + 2x2 1 + c c)2 ln ∣ e 2x + 1 ∣ + c1d)2 ln ∣ ∣e 2x − 1 ∣ + c e) ln |sin x| + c f) − 1 2 ln ∣ ∣x −2 + 4 ∣ + c2. a) ln ∣ ∣ x3 ∣ ∣ + c = 3 ln |x| + c b) ln∣ ∣e sinx ∣ ∣ + c = sin x + cc) − ln ∣ ∣ ∣x−4 ∣ + c = 4 ln |x| + c d) ln ∣x 1/2 1 + c = ln |x| + c2e) ln ∣ ∣ e5x 2+ c = 5x + c f) −3 ln ∣ ∣ x−3/2 + c = ln |x| + cwww.mathcentre.ac.uk 4 c○mathcentre2009
Change language