Indefinite IntegralsDifferentiation allows us to gain a lot of information about a given function. Just as we canthink of having a function f(x) and looking for its derivative, we can also consider goingbackwards - given a function f(x) we would like to find a function F (x) so that F0(x) = f(x).Since this is the reverse process of differentiation, we call it antidifferentiation, and say thatF is an antiderivative of f.Definition: AntiderivativeA function F (x) is an antiderivative of f(x) ifF0(x) = f(x)for all x in the domain of f.We have seen many applications of differentiation, but a natural question to ask is why weare interested in antidifferentiation. One of the primary applications of antidifferentiation isin solving differential equations. Differential equations are equations that involve functionsand their derivatives. Differential equations tend to govern most physical laws, so it isextremely worthwhile to be able to solve them. For instance, Newton’s second law statesthat an object’s mass times its acceleration is given by the sum of the forces acting on it(supposing that the mass of the object is constant), writtenF = maIf we think of acceleration as the second derivative of position, then we have the differentialequationF = md2xdx2In classical mechanics we are interested in solving this differential equation for different phys-ical situations. The solution to the ab ove differential equation depends on what forces areat work in a given situation.We have previously noted that when we differentiate a function, we lose any constant thatis added to the function. Thus, if we have a single antiderivative to a given function, we canfind another antiderivative by adding a constant to it. In fact, for F (x) an antiderivativeof a function f(x), it turns out that every other antiderivative of f(x) can be written asF (x) + c for some constant c. Because we can choose any constant value for c it followsthat this family of antiderivatives contains infinite members. This realization motivates thefollowing definition.Definition: Indefinite IntegralWe call the set of all antiderivatives of f the indefinite integral of f, denoted byZf(x)dxThe symbolRis an integral sign. The function f is the called the integrand and x isthe variable of integration. We say that dx is a differential of x.Based on our previous discussion we can say thatZf(x)dx = F (x) + cbecause the expression on the right-hand side represents all possible antiderivatives of f(x),if we let c be an arbitrary constant.When we find the indefinite integral of a function f(x) we say that we integrate the integrandf(x). Thus, the process of evaluating an integral is referred to as integration. Unfortunately,the notation we have introduced for the indefine integral probably looks rather arcane. Aswe move further into the study of integration, the reason for the notation will become moreclear. The resemblence ofRand an enlongated S is not accidental. This notation reflects therelationship between integration and summation. Integration is essentially the summation ofthe area underneath the integrand f(x) over intervals of infintessimal length. The differen-tial dx represents an infintessimal change in x , which represents the intervals of infintessimallength over which the summation involved in integration occurs. The reason we have thenotation dx is historical - calculus was initially conceived using infintessimals rather thanlimits. However, infintessimals at the time were not mathematically rigorous, but not westill study calculus using limits, because of the vast number of applications of limits (whichwe have yet begun to scratch the surface of). Finally, it is worth noting the resemblencein this dx and the derivative dy/dx. In fact, this notation is also historical, in consideringan infintessimal change in y divided by an infintessimal change in x (this is analagous toconsidering the change in y over an interval as the length h in the limit as h → 0).When it will not lead to confusion we will refer to the indefinite integral as simply the inte-gral. The term indefinite integral is used to distinguish the process of indefinite and definiteintegration. Although both of these concepts are related to the area underneath a function,the indefinite integral is a function, whereas the definite integral is a constant, which is givenby the area underneath a function over a set interval (defined by the limits of integration,which are not present in an indefinite integral).Since the process of (indefinite) integration is an inverse to differentiation, we can derivemany rules for integration using rules we already know for differentiation. For instanceddxxn+1= (n + 1)xnso we see thatddxxn+1n + 1= xnforx 6= −1This leads us to the product rule for integrals.Power Rule for IntegralsZxndx =xn+1n + 1+ cfor every n 6= −1 (n ∈ R \ {−1}).Example 1 Evaluate the indefinite integral of x2.Solution In this case we use the product rule, to see thatZx2=x2+12 + 1+ c =x33+ cExample 2 Evaluate the indefinite integral of√x.Solution Once we rewrite√x = x1/2we see thatZx1/2=x1/2+11/2 + 1+ c =23x3/2+ cExample 3 Evaluate the indefinite integral of x−3.Solution We find thatZx−3=x−3+1−3 + 1+ c = −12x−2+ cExample 4 EvaluateRdt.Solution Rewriting the integrand we findZdt =Zt0dt =t0+10 + 1+ c = t + cJust like differentiation, integration is a linear operation. What this means is that integrationsatisfies the following two properties.Sum Rule for IntegralsZf(x) + g(x)dx =Zf(x)dx +Zg(x)dxConstant Product Rule for IntegralsZaf(x)dx = aZf(x)dxfor every a ∈ RExample 5 Evaluate the indefinite integral of −2x−3+ 4x1/2.Solution We can use the constant product rule and sum rules in conjunction to findZ−2x−3+ 4x1/2= −2Zx−3+ 4Zx1/2= −2(−12x−2) + 4(23x3/2) + c = x−2+83x3/2+ cIn the above analysis we do not write the result as −2c + 4d, because we can easily enoughchoose c and d so that we have any value for the arbitrary constant. Thus, it is cleanerto just replace −2c + 4d with a single c, as both are capable of representing any arbitraryconstant.Corresponding to other differentiation rules we have learned, we can evaluate the followingintegrals.Zexdx = ex+ cZcos(x)dx = sin(x) + cZsin(x)dx = −cos(x) + cNote that the negative sign corresponds to the integral of the sine function, just as thenegative sign corresponds to the derivative of cosine. We’d also like to noteZ1xdx = ln(|x|) + cIn the above