Antiderivative

An antiderivative, also called a primitive, as its name implies, is the opposite of a derivative in calculus. That is, it is a function for which the given function is the derivative. It is important to note that there are an infinite number of antiderivatives for every function, since constants disappear during differentiation. For this reason, an arbitrary constant is often attached to the antiderivative, making an indefinite integral.

The process of antidifferentiation is called indefinite integration or just integration because it uses the integral symbol $\int$. The integral symbol is also used for a closely related operation called definite integration.

Finding an antiderivative

Antidifferentiation is generally much harder than differentiation. The more difficult integrals require some creativity and intuition to solve quickly, and some expressions are impossible to integrate algebraically.

However, some rules for differentiation can be used to make the task easier. For example, the constant multiple rule, i.e. $c \frac{{d}}{{dx}}(x) = \frac{{d}}{{dx}}(cx)$, allows us to easily antidifferentiate power functions. Using 5x³ as an example:

$\frac{{d}}{{dx}}(x^4) = 4x^3$

$\frac{{5}}{{4}}\frac{{d}}{{dx}}(x^4) = \frac{{5}}{{4}}(4x^3)$

$\frac{{d}}{{dx}}\left(\frac{{5}}{{4}}x^4\right) = 5x^3$

Antiderivatives of key general functions

When we differentiate a function, we lose any constant term added to it:

$\frac{d}{dx}(f(x) + c) = \frac{d}{dx}f(x)$ for any constant c

So when we antidifferentiate, we get not one function but a family of functions that differ by a constant. We denote this by writing "+ C" at the end of an expression. This is a sort of in-joke among calculus students — add the + C or you'll get yelled at. (In fact, there is a proof that 0 = 1 based on forgetting the + C.)

What follows are some antiderivatives of common types of functions.

$\int x^n \,d x = \frac{{x^{{n+1}}}}{{n + 1}} + C (n \neq -1)$

$\int \,d x = x + C$

$\int \frac{{\,d x}}{{x}} = \ln |x| +C$

$\int a^x \,d x = \frac{{a^x}}{{\ln a}} + C$

$\int e^x\,dx = e^x + C$