5.1 Antiderivatives and Indefinite IntegrationObjectives1. Write the general solution of a differential equation2. Use indefinite integral notation for antiderivatives3. Use basic integration rules to find antiderivatives4. Find a particular solution of a differential equationSo far, we have talked about finding rates of change in a function.But suppose that we aretrying to go the other way. We know that velocity is the derivative of position. So if we are given thevelocity of an object, are we able to find the position function it came from? This is the idea lurkingbehind antidifferentiation and differential equations: Can we find the original function, given itsrate of change (derivative)?Example: Find a functionFwhose derivative isf(x) = 4x3Definition:A functionFis theantiderivativeoffon an intervalIifFprime(x) =f(x) for allxinI.Now, finding antiderivatives is a little different than most mathematical procedures, since thereisn’toneantiderivative, but rather a wholefamilyof antiderivatives. For example, another an-tiderivative off(x) = 4x3isF(x) =x4-2007.2.However, we are fortunate, because all of thedifferent antiderivatives have the same form:Theorem:IfFis an antiderivative offonI, thenGis also an antiderivative offif and onlyifGhas the formF(x) +Cfor some constantCon the entire intervalI.The constantCis called theconstant of integration, and the family of functions representedby using the ”+C” is thegeneral antiderivativeoff.Example: Find and express the family of antiderivatives forf(x) = sinxNote:We can always check our antiderivative by taking the derivative of it to see if we get theoriginal function.Differential Equations:A differential equation inxandyis an equaiton that involvesx,y,and derivatives ofy. Tosolvea differential equation means to find the functionythat satisfies therelationship given in the equation.Example: Solve the differential equationyprime= 3x

Notationally, this is easier to do in Leibnitz notation:dydx= 3xIn fact, it is easier, usually to ”separate” variables, which means ”multiply” both sides by thedx. This gives:dy= 3xdxNow, the process requires us to antidifferentiate both sides. When we antidifferentiatedyweshould just gety. When we antidifferentiate 3xdxwe want to indicate this process, and the waywe do this is by using the integration symbol:integraldisplaySoy=integraldisplay3xdxIn this expression, 3xis theintegrand, and thedxtells us thevariable of integration- thatis, it tells us what letter is the variable for the problem. In this case we would say we are integratingwith respect tox.Integratingis another way of sayingantidifferentiating. We call the processof finding the antiderivativeindefinite integration, because we haven’t defined the interval overwhich we are finding the antiderivative.Now, to findintegraltext3xdx, we need to know about derivatives.In fact, since we have derivative