Introduction to Antiderivatives
Many processes, both in mathematics and in nature, involve addition. You are familiar with the discrete process of addition, in which you add finitely many numbers to obtain a sum or aggregate. But there are important instances in which we wish to add infinitely many terms. One important example is in the calculation of area—especially the area of an unusual (non-rectilinear) shape. A standard strategy is to approximate the desired area by the sum of small, thin rectangular regions (whose areas are easy to calculate). A second example is the calculation of work, in which we think of the work performed over an interval or curve as the aggregate of small increments of work performed over very short intervals. We need a mathematical formalism for making such summation processes natural and comfortable.
The Concept of Antiderivative
Let f be a given function. We have already seen in the theory of falling bodies (Section 3.4) that it can be useful to find a function F such that F′ = f. We call such a function F an antiderivative of f. In fact we often want to find the most general function F, or a family of functions, whose derivative equals f. We can sometimes achieve this goal by a process of organized guessing.
Suppose that f (x) = cos x. If we want to guess an antiderivative, then we are certainly not going to try a polynomial. For if we differentiate a polynomial then we get another polynomial. So that will not do the job. For similar reasons we are not going to guess a logarithm or an exponential. In fact the way that we get a trigonometric function through differentiation is by differentiating another trigonometric function. What trigonometric function, when differentiated, gives cos x? There are only six functions to try, and a moment’s thought reveals that F(x) = sin x does the trick. In fact an even better answer is F (x) = sin x + C. The constant differentiates to 0, so F′(x) = f(x) = cos x. We have seen in our study of falling bodies that the additive constant gives us a certain amount of flexibility in solving problems.
Now suppose that f(x) = x2. We have already noted that the way to get a polynomial through differentiation is to differentiate another polynomial. Since differentiation reduces the degree of the polynomial by 1, it is natural to guess that the F we seek is a polynomial of degree 3. What about F(x) = x3? We calculate that F′ (x) = 3x2. That does not quite work. We seek x2 for our derivative, but we got 3 x 2. This result suggests adjusting our guess. We instead try F(x) = x3 /3. Then, indeed, F′ (x) = 3x2 /3 = x2 , as desired. We will write F ( x ) = x3 /3 + C for our antiderivative.
More generally, suppose that f ( x ) = ax3 + bx2 + cx + d . Using the reasoning in the last paragraph, we may find fairly easily that F (x) = ax4 /4 + bx3 /3 + cx2 /2 + dx + e . Notice that, once again, we have thrown in an additive constant.
You Try It: Find a family of antiderivatives for the function f (x) = sin 2x − x4 + ex.
Find practice problems and solutions for these concepts at: The Integral Practice Test.
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