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 £© Section
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 £© Using the Fundamental Theorem
As we saw in Section , using the Fundamental Theorem of Integral Calculus reduces
the problem of evaluating a definite integral to the problem of finding an antiderivative.
Unfortunately, finding antiderivatives, even for relatively simple functions, cannot be done
as routinely as putation of derivatives. For example, suppose we let f(x) = sin(x),
g(x) = x, and
f(x) sin(x)
h(x) = = .
g(x) x
Then, since we know the derivative of f and we know the derivative of g, it is a simple
matter to find the derivative of h using the quotient rule. However, knowing the an-
tiderivatives of f and g in no way helps us find the antiderivative of h. In fact, it has been
shown that the antiderivative of h is not expressible in terms of any bination of
algebraic and elementary transcendental functions. Because of results like this, many of
the definite integrals that are encountered in applications cannot be evaluated using the
Fundamental Theorem of Integral Calculus; instead, they must be approximated using nu-
merical techniques such as those we studied in Section . Of course, when antiderivatives
are available, the Fundamental Theorem is the best way to evaluate an integral. To this
end, we will investigate, in this section and in the next, techniques for evaluating definite
integrals by finding antiderivatives and applying the Fundamental Theorem.
Before we begin, we need to introduce some additional notation and terminology.
First of all, we will call the collection of all antiderivatives of a given function f the general
antiderivative of f. For example, if f(x) = 3x2, then the general antiderivative of f is
given by F (x) = x3 + c, where c is an arbitrary constant.
Second, since the Fundamental Theorem of Calculus draws a close connection between
antiderivatives and definite integrals, it is customary to borrow the