An antiderivative for a function \(f(x)\) is a function \(F(x)\) such that \(F'(x) = f(x)\). In this sense the antiderivative is an inverse process to the derivative.
An example: Since \((\sin(3x))' = 3\cos(3x)\), we can say that the function \(\sin(3x)\) is an antiderivative for the function \(3\cos(3x)\).
Note that also \(\sin(3x) + 5\) would be an antiderivative, as an added constant goes away when we differentiate. In that sense we can have many antiderivatives for a function. They are however all related to each other via adding a constant, as the following theorem describes:
Theorem
If \(F(x)\) is an antiderivative of \(f(x)\) on an interval \((a, b)\), then all the antiderivatives of \(f(x)\) have the form \(F(x) + C\) for some constant \(C\).
We use the symbol \(\int f(x)dx\) to denote these antiderivatives, and we call it the indefinite integral. Equalities involving indefinite integrals only make sense up to an additive constant.
One standard example where we can compute the antiderivative is polynomials. This essentially reverses the power rule that \(\left(x^{n+1}\right)' = (n+1)x^n\):
Power Rule for Antiderivatives
\[\int x^n dx = \frac{x^{n+1}}{n+1} + C\] as long as \(n\neq 1\).
The antiderivative of \(\frac{1}{x}\) cannot be obtained via the above rule. It turns out to be a very interesting function, called the natural logarithm, and it will be defined and discussed more in Calculus 2.
The above theorem also shows us a general process for figuring out some of these antiderivatives: If we can guess a function that has the desired derivative, then we have found our antiderivative. For instance this way of thinking can produce the following antiderivatives:
Basic Trigonometric Antiderivatives
\[\int\sin x dx = -\cos x + C\] \[\int\cos x dx = \sin x + C\] \[\int\sec^2 x dx = \tan x + C\] \[\int\sec x\tan xdx = \sec x + C\]
A first application of these ideas is the solution of simple differential equations. A differential equation is an equation where the unknown is a function \(y=y(x)\), and the equation involves the derivatives of the function. For example the antiderivative of a function \(f(x)\) solves the differential equation: \[\frac{dy}{dx} = f(x)\]
In these situations the solution is only determined up to a constant \(C\). The constant can often be determined if they also provide us the value that the function must take at a particular point. This is often called an initial value.
Example: Find the function \(y\) such that \(\frac{dy}{dx} = x^3\) and with the initial value \(y(0) = 3\).
We would start by computing \(\int x^3dx = \frac{1}{4}x^4 + C\). So this tells us that our function \(y\) must have the form \(y(x) = \frac{1}{4} x^4 + C\). We just need to find the \(C\).
But they also told us that \(y(0) = 3\). This must mean that \(3=\frac{1}{4}\times 0^4 + C = C\), so the constant must be \(C=3\). So we have our final answer: \[y(x) = \frac{1}{4}x^4 + 3\]