Chapter 13 – 2a Integration by Substitution



Chapter 13 – 2a Integration by SubstitutionReversing the Chain RuleChain Rule: If u=g(x) then df(u)dx=df(u)dududxThe corresponding integration rule is given by:dfudududxdx=fu+C=fgx+CThis is sometimes written as:f'udu=fu+C=fgx+CSome General Indefinite Integral Formulas Based on the Chain Rulefxnf'(x)dx=1n+1fxn+1+C=fxn+1n+1+C, n≠-1fx-1f'(x)dx=1fxf'(x)dx=lnfx+Cefxf'(x)dx=efx+CExample 1. Find the indefinite integral of fx=3x+523u=3x+5, du=3dx3x+523dx=u2du=u33+C=3x+533+CExample 2. Find the indefinite integral of fx=5x3+1-315x2u=5x3+1, du=15x2dx5x3+1-315x2dx=u-3du=u-2-2+C=-u-22+C=-5x3+1-22+CExample 3. Find the indefinite integral of fx=5e5xu=5x, du=5dx5e5xdx=e5x5dx=eudu=eu+C=e5x+CExample 4. Find the indefinite integral of fx=x+310u=x+3, du=dxx+310dx=u10du=u1111+C=x+31111+CExample 5. Find the indefinite integral of ft=6t-7-2u=6t-7, du=6dt6t-7-2dt=166t-7-26dt=16u-2du=16-u-1+C=-166t-7-1+CExample 6. Find the indefinite integral of fx=x24-x34u=4-x3, du=-3x2dxx24-x34dx=-13-3x24-x34dx=-13u-4du=-13-u-33+C=194-x3-3+CExample 7. Find the indefinite integral of fx=xx+4u=x+4, x=u-4, and du=dxxx+4dx=xx+412dx=u-4u12du=u32-4u12du=u5252-4u3232+C=25x+452-83x+432+CExample 8. Find the indefinite integral of fx=xx-49u=x-4, x=u+4, and du=dxxx-49dx=u+4u9du=u10+4u9du=u1111+4u1010+C=x-41111+2x-4105+CExample 9. The marginal price for a weekly demand of x bottles of baby shampoo in a drugstore is given byp'x=-6,0003x+502Find the price-demand equation if the weekly demand is 150 when the price is $4.00 a bottle. Estimate the weekly demand if the price falls to $2.50 a bottle.u=3x+50, du=3dxp=-6,0003x+502dx=-2,0003x+50-23dx=-2,000u-2du=-2,000u-1-1+C=2,0003x+50+C425958039370p150=2,0003?150+50+C=4.00 ? C=0 and p(x)=2,0003x+50Demand: x(p)=132,000p-50x(2.50)=132,0002.50-50=250 bottles ................
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