14.2 Integration in practice
Even though we understand what an integral is, we are far from computing them in practice. As opposed to differentiation, analytically evaluating integrals can be really difficult and sometimes downright impossible. The formula (92) suggests that the key is to find the function whose derivative is the integrand, called the antiderivative or primitive function. This is harder than you think. Nevertheless, there are several tools for this, and we are going to devote this section to studying the most important ones.
Often, the key is finding the antiderivative, so we introduce the notation
for the functions where F′ = f. (Sometimes we abbreviate this to F = ∫ fdx.) Note that since (F + some constant)′ = F′, the antiderivative ∫ f(x)dx is not uniquely determined. However, this is not an issue for us, as the Newton-Leibniz formula states that
Thus, any additional constants would be eliminated.
With this under our belt,...
