Integral Antiderivatives and Properties Overview

f(x)dx= F(x) + C xn+1 n + 1 + C for all n = 1 n x dx = 1 xdx= ln |x|+ C (Note the absolute values) ex dx = ex + C ax lna (don t memorize: use ax = ex ln a) x a dx= + C sinxdx = cosx+ C cosxdx= sinx+ C tan xdx= ln |cosx| + C (Note the absolute values) 1 1 + x2dx = arctanx+ C 1 1 x2dx= arcsin x+ C The following integral is also useful, but not as important as the ones above: dx cosx 1 2 1+ sinx 1 sinx 2 2 + C for = ln <x< . Table 1. The list of the standard integrals everyone should know Table1lists a number of antiderivatives which you should know. All of these integrals should be familiar from the differentiation rules we have learned so far, except for for the integrals of tan x and of1. You a b can check those by differentiation (using ln cos x = ln a ln b simplifies things a bit). 6. Properties of the Integral Just as we had a list of properties for the limits and derivatives of sums and products of functions, the integral has similar properties. Suppose we have two functions f (x) and g(x) with antiderivatives F (x) and tt(x), respectively. Then we know that d dx in words, F + tt is an antiderivative of f + g, which we can write as (51) . d j Similarly, cF (x) = cF (x) = cf(x) implies that (52) cf(x)dx= c . j j F(x) + tt(x) = F (x) + tt(x) = f(x) + g(x), .f (x) + g(x) dx= f(x)dx+ g(x)dx. dx f(x)dx if c is a constant. 1He took math 222 and learned to integrate by parts. 2

These properties imply analogous properties for the definite integral. For any pair of functions on an interval [a, b] one has (53) f(x) + g(x) dx= a and for any function f and constant c one has (54) cf(x)dx = c a b b b f(x)dx+ g(x)dx, a a b b f(x)dx. a Definite integrals have one other property for which there is no analog in indefinite integrals: if you split the interval of integration into two parts, then the integral over the whole is the sum of the integrals over the parts. The following theorem says it more precisely. 6.1. Theorem. Given a < c < b, and a function on the interval [a, b]then b c b f(x)dx= f(x)dx+ f(x)dx. (55) a a c Proof. Let F be an antiderivative of f . Then c b f(x)dx= F(c) F(a) and f(x)dx= F(b) F(a), a c so that b f(x)dx= F(b) F(a) a = F(b) F(c) + F(c) F(a) c = f(x)dx+ a b f(x)dx. c Q b a So far we have always assumed theat a < b in all indefinitie integrals suggests that when b < a, we should define the integral as b (56) f(x)dx= F(b) F(a) = F(a) F(b) = . . .. The fundamental theorem . a f(x)dx. a b For instance, 0 1 1 2 xdx= xdx = . 1 0 7. The deftnite integral as a function of its integration bounds Consider the expression x t2dt. I = 0 What does I depend on? To see this, you calculate the integral and you find 1 3 3 x 1 3 1 3 1 3 = x 0 = x. I = t 3 3 3 0 So the integral depends on x. It does not depend on t, since t is a dummy variable (see 3.3where we already discussed this point.) In this way you can use integrals to define new functions. For instance, we could define x t2dt, I(x) = 0 3

which would be a roundabout way of defining the function I(x) = x3/3. Again, since t is a dummy variable we can replace it by any other variable we like. Thus I(x) = x 2d 0 defines the same function (namely, I(x) = 1x3). 3 The previous example does not define a new function (I(x) = x3/3). An example of a new function defined by an integral is the error-function from statistics. It is given by x 2 def t2 e erf(x) = (57) dt, 0 so erf(x) is the area of the shaded region in figure3. The integral in (57) cannot be computed in terms of the 2 2 x y= e Area = erf(x) x Figure 3. Definition of the Error function. standard functions (square and higher roots, sine, cosine, exponential and logarithms). Since the integral in (57) occurs very often in statistics (in relation with the so-called normal distribution) it has been given a name, namely, erf(x) . How do you differentiate a function that is defined by an integral? The answer is simple, for if f (x) = Fj(x) then the fundamental theorem says that x f(t) dt= F(x) F(a), a and therefore d dx a i.e. d dx a A similar calculation gives you d dx x So what is the derivative of the error function? We have , d erf (x) = dx 2 d = dx 2 = e d . x j F(x) F(a) = F (x) = f(x), dx f(t) dt= x f(t) dt= f(x). b f(t) dt= f(x). , x 2 t2dt j e 0 x t2 e dt 0 2 x . 4