The indefinite integral

Lately we have seen how the definite integral arises in many different places and so it is natural to want to compute definite integrals effectively. Remember that the Fundamental Theorem of Calculus gives us a very powerful tool for doing so: it says that if we want to compute the definite integral $ \int_a^b f(x) ~dx $ , then we only need to find an antiderivative G of f . Then

\[ \int_a^b f(x) ~dx = G(b) - G(a) \]

While we have been able to write down antiderivatives for some simple functions, there are still many more complicated functions whose antiderivatives are difficult for us to find right now. In this section, we will learn a few tecniques for finding antiderivatives.

To this end, we want to introduce one piece of notation. By the indefinite integral of a function f, we mean the general antiderivative of f. We will denote the indefinite integral by

\[ \int f(x)~dx \]

For example, if $ n \neq -1 $ , we have

\[ \int x^n~dx = \frac{x^{n+1}}{n+1} + C \]

When we write this, C will denote an arbitrary constant. This is because the function f has many antiderivatives, but any two differ by a constant. Consequently, if we find one antiderivative and add on an arbitrary constant, we have written every antiderivative.

Here are some more examples:

\[ \begin{array}{c} \int 1 ~dx = x + C \\ \\ \int x ~dx = \frac{x^2}{2} + C \\ \\ \int x^n ~dx = \frac{x^{n+1}}{n+1} +C \mbox{ if } n\neq -1 \\ \\ \int \frac 1x ~dx = \ln |x| + C \\ \\ \int \sin(x) ~dx = -\cos(x) + C \\ \\ \int \cos(x) ~dx = \sin(x) + C \\ \\ \int e^{x} ~dx = e^{x} + C \\ \\ \int \frac{1}{1+x^2} ~dx = \tan^{-1}(x) + C \\ \\ \int \frac{1}{\sqrt{1-x^2}} = \sin^{-1}(x) + C \end{array} \]