Page 113 - Calculus Demystified

The Integral
CHAPTER 4
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such a function F an antiderivative of f . In fact we often want to find the most
general function F,ora 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.
2
Now suppose that f(x) = x . 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
3
the F we seek is a polynomial of degree 3. What about F(x) = x ? We calculate
2
2

that F (x) = 3x . That does not quite work. We seek x for our derivative, but
2
3
we got 3x . This result suggests adjusting our guess. We instead try F(x) = x /3.
2
2
3
Then, indeed, F (x) = 3x /3 = x , as desired. We will write F(x) = x /3 + C

for our antiderivative.
2
3
More generally, suppose that f(x) = ax + bx + cx + d. Using the reasoning
4 3
in the last paragraph, we may find fairly easily that F(x) = ax /4 + bx /3 +
2
cx /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 −
x
4
x + e .
4.1.2 THE INDEFINITE INTEGRAL
In practice, it is useful to have a compact notation for the antiderivative. What we
do, instead of saying that “the antiderivative of f(x) is F(x) + C,” is to write

f(x) dx = F(x) + C.
So, for example,

cos xdx = sin x + C
and
4 2
3 x x
x + xdx = + + C
4 2