Math 1A: introduction to functions and calculus Oliver Knill, 2011Lecture 29: Integration by partsIf we integrate the product rule (uv)′= u′v + uv′we obtain an integration rule called integrationby parts. It is a powerful tool, which complements substitution. As a rule of thumb, always tryfirst to simplify a function and integrate directly, then give substitution a first shot before tryingintegration by parts.Ru(x) v’ (x)dx = u(x)v(x) −Ru′(x)v(x) dx.1 FindRx sin(x) dx. Solution. L ets identify the part which we want to differentiate andcall it u and the part to integrate and call it v′. The integration by pa rts method nowproceeds by writing down uv and subtracting a new integral which integrates u′v :Zx sin(x) dx = x (−cos(x)) −Z1 (− cos(x) ) dx = −x cos(x) + sin(x) + C dx .2 FindRxexdx. Solution.Zx exp(x) dx = x exp(x) −Z1 exp(x) dx = x exp(x) − exp(x) + C dx .3 FindRlog(x) dx. Solution. There is only one function here, but we can look at it aslog(x) · 1Zlog(x) 1 dx = log(x)x −Z1xx dx = x log(x) − x + C .4 FindRx log(x) dx. Solution. Since we know from the previous problem how to integratelog we could proceed like t his. We would get through but what if we do not know? Letsdifferentiate log(x) and integrate x:Zlog(x) x dx = log(x)x22−Z1xx22dxwhich is log(x)x2/2 − x2/4.We see that it is better to differentiate log first.5 Marry go round: Find I =Rsin(x) exp(x) dx. Solution. Lets integrate exp(x) anddifferentiate sin(x).= sin(x) exp(x) −Zcos(x) exp(x) dx .Lets do it again:= sin(x) exp(x) − cos(x) exp(x) −Zsin(x) exp(x) dx.We moved in circles and are stuck! Are we really. We have derived an identityI = sin(x) exp(x) − cos(x) exp(x) − Iwhich we can solve for I and getI = [sin(x) exp(x) − cos(x) exp(x)]/2 .12Tic-Tac-ToeIntegration by part s can bog you down if you do it sev-eral times. Keeping the order of the signs can be daunt-ing. This is why a tabular integration by partsmethod is so powerful. It has been called ”Tic-Tac-Toe” in the movie Stand and deliver. Lets call it Tic-Tac-Toe therefore.6 Find the anti-derivative of x2sin(x). Solution:x2sin(x)2x −cos(x) ⊕2 −sin(x) ⊖0 cos(x) ⊕The antiderivative is−x2cos(x) + 2x sin(x) + 2 cos(x) + C .7 Find the anti-derivative of (x − 1)3e2x. Solution:(x − 1)3exp(2x)3(x − 1)2exp(2x)/2 ⊕6(x − 1) exp(2x)/4 ⊖6 exp(2x)/8 ⊕0 exp(2x)/16 ⊖The anti-derivative is(x − 1)3e2x/2 − 3 ( x − 1)2e2x/4 + 6(x − 1)e2x/8 − 6e2x/16 + C .8 Find the anti-derivative of x2cos(x). Solution:x2cos(x)2x sin(x) ⊕2 −cos(x) ⊖0 −sin(x) ⊕The anti-derivative isx2sin(x) + 2x cos(x) − 2 sin(x) + C .Ok, we are now ready for more extreme stuff.39 Find the anti-derivative of x7cos(x). Solution:x7cos(x)7x6sin(x) ⊕42x5−cos(x) ⊖120x4−sin(x) ⊕840x3cos(x) ⊖2520x2sin(x) ⊕5040x −cos(x) ⊖5040 −sin(x) ⊕0 cos(x) ⊖The anti-derivative isF (x) = x7sin(x)+ 7x6cos(x)− 42x5sin(x)− 210x4cos(x)+ 840x3sin(x)+ 2520x2cos(x)− 5040x sin(x)− 5040 cos(x) + C .Do this without this method and you see the value of the method.1 2 3:I myself learned the method from the movie ”Stand andDeliver”, where Jaime Escalante of the Garfield HighSchool in LA uses the method. It can be traced downto an ar ticle of V.N. Murty. The method realizes in aclever way an iterated integration by parts method:Zfgdx = f g(−1)− f(1)g−2+ f(2)g(−3)− . . .− (−1)nZf(n+1)g(−n−1)dxwhich can easily shown to be true by induction and jus-tifies the method: the f function is differentiat ed againand again and the g function is integrated again andagain. You see, where the alternating minus signs comefrom. You see that we always pair a k’th derivative witha k + 1’th integral and take the sign (−1)k.Coffee or Tea?1V.N. Murty, Integration by parts, Two-Year College Mathematics Journal 11, 1980, pages 90-94.2David Horowitz, Tabular Integration by Parts, College Mathematics Journal, 21, 1990, pages 307-311.3K.W. Folley, integration by parts, American Mathematical Monthly 54, 1947, pages 542-5434When doing integration by parts, We want to try first to differentiate Logs, Inverse trig functions,Powers, Trig functions and Exponentials. This can be remembered as LIPTE which is close to”lipton” (the tea).For coffee lovers, there is an equivalent one: Logs, Inverse t rig functions, Algebraic functions,Trig functions and Exponentials which can be remembered as LIATE which is close to ”lat t e”(the coffee).Whether you pr efer to remember it as a ”coffee latte” or a ”lipton tea” is up to you.There is even a better method, the ”opportunistic method”:Just integrate what you can integrate and differentiate the rest.An don’t forg et to consider integrating 1, if nothing else works.Homework1 Integr ateRx2log(x) dx.2 IntegrateRx5sin(x) dx3 Find the anti derivative ofRx6exp(x) dx. (*)4 Find the anti derivative ofR√x log(x) dx.5 Find the anti derivative ofRsin(x) exp(−x) dx.(*) If you want to go for the record. Lets see who can integrate the largest xnexp(x)! It has tobe done by hand, not with a computer algebra system
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