Fundamentals
This section introduces the concept of antidifferentiation and linearity.
Definition
Antiderivatives are the inverse of the differentiation. Given some function f(x) where
f^\prime(x) = g(x)
the antiderivative of g(x) is f(x). In general, the antiderivative of a function g(x) is a separate function f(x) whose we must differentiate to obtain g(x). This yields our definition.
These antiderivatives are also called indefinite integrals, or just integrals for short. For any function g(x), there exists an infinite set of antiderivatives f(x) which only vary by some constant. Thus we write
\int g(x) \, dx = f(x) + C
where f(x) is some antiderivative of g(x) and C is any real number. Antiderivatives do not always exists for any arbitrary function g(x), and when they do exist, they may not be possible to express in elementary terms.
From this point on, we will refer almost exclusively to antiderivatives as integrals. We will call functions integrable if their antiderivative exists.
Linearity
Integrals have a property called linearity. This means they are closed under linear transformations of functions like addition and scaler multiplication. For example
\int f(x) + g(x) \,dx = \int f(x) \,dx + \int g(x) \,dx.
Or, given some constant C
\int C f(x) \,dx = C \int f(x) \,dx.
This is presented in the following theorem without proof.
Problems
For all problems, let F^\prime(x) = f(x) and G^\prime(x) = g(x),