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Math 19BMidterm 2Study AidNovember 16, 2010Pre-Chapter 7.1 - Stuffyou Should Know!•Basic Antiderivatives.The following is the minimum list of antiderivatives you should have eithermemorized or written on your review card (Candcare constants):c dx=cx+Cxndx=xn+1n+ 1+C(n=−1)1xdx= ln|x|+Cexdx=ex+Csinx dx=−cosx+Ccosx dx= sinx+Csec2x dx= tanx+Cf(x)f(x)dx= ln|f(x)|+C1x2+a2dx=1aarctanxa+C1√1−x2dx= arcsinx+C•U-substitution: For indefinite integrals theu-subs formula may be expressed:f(g(x))g(x)dx=f(u)duwhereu=g(x). In the definite case the formula reads:baf(g(x))g(x)dx=g(b)g(a)f(u)du.Chapter 7.1 - Integration by parts.Formula.•Integration by parts formula.f(x)g(x)dx=f(x)g(x)−g(x)f(x)dxor, in the more catchy form:u dv=uv−v du.1
Remarks.•ILATE.When considering the product of two functions in an integration by parts problem, we need tochoose one function to beuand one to bedv. For your ‘first attempt’ chooseuby ‘order of appearance’in the ‘ILATE’ sequence:I−Inverses (trig inverses such as arcsinxetc.)L−Logarithms (usually lnx)A−Apolynomial (eitherx,x2orx3.)T−Trig function (sin(x), cos(2x), etc.)E=Exponential (ex,e3x, etc.).•Multiply by 1.Remember you can writef(x) as 1×f(x). This trick is commonly used in the ‘I’ or‘L’ cases of ILATE; notably in calculatinglnx dxorarcsinx dx.•Multiple integration by parts.If you chooseu=x2in the ‘A’ case of ILATE you may have to dointegration by parts (IBP) twice. Similarly, if you’re forced to chooseu=x3, you may have to do IBPthree times!In the ‘T’ or ‘E’ cases of ILATE you’ll probably have do IBPs twice and use the trick that the integralyou get the second time around is exactly what you started with.If you haven’t seen this type ofexample or have no idea what I’m going on about (!) try:e4xsin(2x)dx.Chapter 7.2 - Trigonometric integrals.Formula.•Here are all the trig identities you should need:sin2x+ cos2x= 1sin2x=12(1−cos(2x))tan2x+ 1 = sec2xcos2x=12(1 + cos(2x))1 + cot2x= csc2xsin(2x) = 2 sinxcosx.Techniques.•sinmxcosnx dx.1. Ifmis odd, factor out one sinxand use sin2+ cos2= 1 to express the rest of the integral in termsof cosx. Then make the u-subs,u= cosx.2. Ifnis odd, factor out one cosxand use sin2+ cos2= 1 to express the rest of the integral in termsof sinx. Then make the u-subs,u= sinx.3. Ifmandnare both even use the half angle (‘power reducing’) identities:sin2x=12(1−cos(2x))andcos2x=12(1 + cos(2x)).•tanmxsecnx dx.Page 2
1. Ifnis even (think sEc =Even), factor out one sec2xand use tan2+1 = sec2xto express therest of the integral in terms of tanx. Then make the u-subs,u= tanx.