Louisiana State University



Section 4.9 AntiderivativesTopic 1: AntiderivativesDefinitionAntiderivativeA function F is an antiderivative of f on an interval I provided F'x=f(x), for all x in I.TheoremThe Family of AntiderivativesLet F be any antiderivative of f on an interval I. Then all the antiderivatives of f on I have the form F+C, where C is an arbitrary ic 2: Indefinite IntegralsThe notation ddx(fx) means take the derivative of f with respect to x. We need analogous notation for antiderivatives. For historical reasons, the notation that means find the antiderivative of f with respect to x is the indefinite integral fx?dx. Every time an indefinite integral sign appears it is followed by a function called the integrand and the differential (dx when x is the independent variable). The notation fx?dx represents all the antiderivatives of f. TheoremPower Rule for Indefinite Integralsxp?dx=xp+1p+1+Cwhere p≠-1 is a real number and C is an arbitrary constant TheoremConstant Multiple and Sum RulesConstant Multiple Rule:cfx?dx=cfx?dxfor real numbers cSum Rule:fx+gx?dx=fx?dx+gx?dx Topic 3: Indefinite Integrals of Trigonometric FunctionsIndefinite Integrals of Trigonometric Functionsddxsin?ax=a cos?ax cos?ax?dx=1asin?ax+Cddxcos?ax=-a sin?ax sin?ax?dx=-1acos?ax+Cddxtan?ax=a sec2ax sec2axdx=1atan?ax+Cddxcot?ax=-a csc2ax csc2axdx=-1acot?ax+Cddxsec?ax=a secax?tanax secax?tanaxdx=1asec?ax+Cddxcsc?ax=-a cscax?cotax cscax?cot?axdx=-1acsc?ax+CTopic 4: Other Indefinite IntegralsOther Indefinite Integrals ddxeax=aeax eax?dx=1aeax+Cddxbx=bxln?b bx?dx=1ln?bbx+C, b>0 and b≠1ddxln?|x|=1x dxx=ln?|x|+Cddx(sin-1xa)=1a2-x2 dxa2-x2=sin-1xa+Cddxtan-1xa =aa2+x2 dxa2+x2=1atan-1xa+Cddx(sec-1xa)=axx2-a2 dxxx2-a2=1asec-1xa+C, a>0Topic 5: Introduction to Differential EquationsAn equation involving an unknown function and its derivative is called a differential equation. In many cases, the differential equation is accompanied by an initial condition that allows us to determine the constant of integration. A differential equation coupled with an initial condition is called an initial value ic 6: Motion Problems RevisitedInitial Value Problems for Velocity and PositionSuppose an object moves along a line with a (known) velocity v(t), for t≥0. Then its position is found by solving the initial value problems't=v(t), s0=s0 where s0 is the initial position.If the acceleration of the object a(t) is given, then its velocity is found by solving the initial value problemv't=a(t), v0=v0 where v0 is the initial velocity. ................
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