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Chapter 5 – Integrals IStandard Integralsax+bn dx=ax+bn+1n+1a+C , (n≠1)1ax+b dx=1alnax+b+Ceax+b dx=1aeax+b+Cbx dx=bxlnb+Csinax+b dx=-1acosax+b+Ccosax+b dx=1asinax+b+Ctan(ax+b) dx=1alnsec(ax+b)+Csecax+b dx=1alnsecax+b+tanax+b+Ccscax+b dx=-1alncscax+b+cotax+b+Ccot(ax+b) dx=-1alncscax+b+Csec2ax+b dx=1atanax+b+Ccsc2(ax+b) dx=-1acotax+b+Csecax+b?tanax+b dx=1asecax+b+Ccsc(ax+b)?cotax+b dx=-1acscax+b+C1a2+x+b2 dx=1atan-1x+ba+C1a2-x+b2 dx=sin-1x+ba+C-1a2-x+b2 dx=cos-1x+ba+C1a2-x+b2 dx=12alnx+b+ax+b-a+C1x+b2-a2 dx=12alnx+b-ax+b+a+C1x+b2+a2 dx=lnx+b+x+b2+a2+C1x+b2-a2 dx=lnx+b+x+b2-a2+CPartial FractionsFactors of V(x)Partial Fractionsax+bAax+bax+b2Aax+b+Bax+b2ax2+bx+cAx+Bax2+bx+c b2-4ac<0a3+(b)3=(a+(b))(a2-a(b)+b2)Integration by Substitutionf(gx)?g'xdx=fu duTrigonometric SubstitutionExpressionSubstitutionDomaina2-x+b2x+b=asinθ-π2≤θ≤π2a2+x+b2x+b=atanθ-π2≤θ≤π2x+b2-a2x+b=asecθ0≤θ<π2 or π≤θ<3π2Integration by Partsf'xgx dx=fx gx-fx g'x dxTypeExamplesRemarkLogarithmiclnax+b dxSubstitute u=ax+b to simplify integralInverse Trigometricsin-1ax+b dx cos-1ax+b dx tan-1ax+b dx AlgebraicPower functions xa, polynomialsTrigometricsinax+b, cosax+b, tanax+b, cscax+b, secax+b, cotax+bExponentialeax+bTrigonometric Identities Useful for Integrationsec2θ-1=tan2θcsc2θ-1=cot2θsinAcosA=12sin2Acos2A=121+cos2Asin2A=121-cos2AsinA?cosB=12[sinA+B+sinA-B]cosA?sinB=12[sinA+B-sinA-B]cosA?cosB=12cosA+B+cosA-BsinA?sinB=-12cos(A+B)-cos(A-B)Chapter 6 – Integrals IIRiemann Sums01fx dx=limn→∞1ni=1nf(in)Force 1n out firstExpress as fin=f(x)abf(x) dx≈limn→∞i=1nb-an?f(a+ib-an)Fundamental Theorem of CalculusFTC 1abf(x) dx=Fb-F(a)FTC 2ddxv(x)u(x)f(t) dt=fux?u'x-fvx?v'(x)Miscellaneouspx+qax2+bx+c dxAf'+Bf dx=Alnf+B1f dxpx+qax2+bx+c dxAf'+Bf dx=A 2f+B1f dxLet fx=ax2+bx+cFind f'(x)Find A and B by fitting f' into px+q0af(x) dx=0af(a-x) dxlimfxgx=explimgx?lnf(x)Examples19117371670409Product Rule + FTC, don’t touch integral part00Product Rule + FTC, don’t touch integral part1728857358444Take out term with x00Take out term with xFTCFind exact value of f'(0)fx=1e2x3+x1+lnt dt=3+x1e2x11+lntdtf'x=3+x?11+2x?2e2x+1e2x11+lnt dtf'0=3?2+1111+lnt dtChapter 7 – Applications of Integrals IArea on x-axisA=abfx-g(x) dxArea on y-axisA=cdfy-g(y) dyDisk MethodV=πabfx2 dxV=πcdfy2 dyShell MethodV=2πabxf(x) dxV=2πabxfx-g(x) dxV=2πcdyf(y) dyV=2πcdyfy-g(y) dyInverse FunctionsUse on Inverse trigo and logsabfx dx=b fb-a fa-fafbf-1x dxsec-1t=cos-11tcsc-1t=sin-11tcot-1t=tan-11tArc Lengthcd1+dydx2 dx–OR–cd1+dxdy2 dyWhen f cont. on c,d, works on f>0 & f<0Chapter 8 – Applications of Integrals IIProbability Density Functionfx≥0 for all xProbability > 0-∞∞f(t) dt=1Total Probability = 1Px1≤X≤x2=x1x2f(t) dtProbability = area under curvePX=x=0?xPX≤x=PX<x=-∞xf(t) dt206049280038ALWAYS P(X≤z)00ALWAYS P(X≤z)Cumulative Distribution FunctionFx=-∞xft dt (=P(X≤x)Expected Value (Mean)EX=abt ft dt E(gX=abgt ft dtValues of a and b depends on questionFirst Order Ordinary Differential EquationsSeparable ODE1g(y) dy=f(x) dxy on one side, x on otherLinear ODEdydx+Px?y=QxIx=expPx dxy?Ix=Qx?Ix dxSet dydx=…Check separability1gy dy=fxdx –OR– dydx+Px?y=Q(x)4a) Integrate both sides4b) Find Ix=expP(x) dx5a) Add arbitrary constant, C 5b) y?Ix=Qx?Ix dx6a) Combine ln terms6b) y?Ix=…+C7a) y=…7b) y=…101981031145500GeometryLaw of SinessinAa=sinBb=sinCcLaw of Cosinesc2=a2+b2-2abcosCArea of triangle12×base×height12absinCs(s-a)(s-b)(s-c)where s=12(a+b+c)SolidVolumeSurface AreaCone13πr2hπrr2+h2Cylinderπr2h2πrhSphere43πr34πr215505042146300Basically just wants to find area under curve in x00Basically just wants to find area under curve in x172402518415Fx=DFx… dt00Fx=DFx… dtleft19878300Cumulative Distribution FunctionRiemann Sumsi=0ni=n2(n+1)i=0ni2=n6(n+1)(2n+1) i=0ni3=n24n+12 ................
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