Given a function \(f(x)\), we'll often need to find a function whose derivative is equal to \(f(x)\).
The function, which has derivative \(f(x)\), is known as the integral of \(f(x)\) and we'll frequently refer to it
using the capital letter \(F(x)\).
In other words: the integral of \(f(x)\) is a function \(F(x)\) such that \(F'(x) = f(x)\).
For instance, given the function \(f(x) = 2x\), the function \(F(x) = x^2\) is an integral of \(f(x)=2x\) since \(F'(x) = 2x\).
In fact any function \(F(x) = x^2 + c \), where \(c\in \mathbb{R}\) will have derivative \(F'(x) = 2x\).
This generic function \(F(x) = x^2 + c \) is known as the antiderivative (sometimes called the primitive) of \(f(x)\).
Given a continuous function \(f(x)\), we define its antiderivative \(F(x)\) as the function whose derivative equals \(f(x)\).
That's:
\[F'(x) = f(x)\]
Note: the antiderivative is usually referred to using the capitalized letter of the letter used for the initial function; \(F(x)\) for \(f(x)\), \(G(x)\) for \(g(x)\), \(H(x)\) for \(h(x)\), ... .
The function \(F(x)\) is also known as the indefinite integral, loosely said "integral", of \(f(x)\) and we write:
\[F(x) = \int f(x) dx \]
Note: this notation will make more sense once we have studied definite integrals, for now it should be accepted and known.
For instance, if \(f(x) = 3x^2\) we define its antiderivative as the function:
\[F(x) = x^3 + c\]
Since \(F'(x)=3x^2\).
We can also write this as the indefinite integral:
\[\int 3x^2 dx = x^3 + c\]
Where \(c\) is known as the constant of integration and should not be forgotten.
No matter the value of \(c \in \mathbb{R}\), when we differentiate \(x^3 + c\): \(c\) will disappear. This shows us
that there is an infinite number of functions that have derivative \(3x^2\), all of which can be written \(F(x) = x^3 + c \).
Try finding an expression for each of the following integrals: