Antidifferentiation

Consider a function \(F\) and its derivative \(f\) (i.e. \(f = F').\) Since \(f\) is the derivative of \(F,\) let’s refer to \(F\) as an antiderivative of \(f.\) Since the operation of “taking the derivative” is referred to as differentiation, let’s refer to “taking an antiderivative” as antidifferentiation.

A function’s derivative is unique, but it’s antiderivative is not. A function has infinitely many antiderivatives, all of them differing by some additive constant \(C.\) To account for this, while the operation of differentiation is denoted either \(F' = f\) or \(\frac{\mathrm{d}}{\mathrm{d}x}F = f,\) let’s denote antidifferentiation this way: \(\int f \,\mathrm{d}x = F + C.\) The tall “S” symbol \(\int\) on the left is called an integral sign, the little \(\mathrm{d}x\) is called the differential, the \(+C\) is called the constant of integration, and altogether \(\int f \,\mathrm{d}x\) is referred to as the indefinite integral of \(f\) with respect to \(x.\)

Formulas

\(\displaystyle \int k \,\mathrm{d}x = kx + C\,\) (for constant \(k\))

\(\displaystyle \int x^n \,\mathrm{d}x = \tfrac{1}{n+1}x^{n+1} + C\,\) (for \(n\!\neq\!-1\))

\(\displaystyle \int \tfrac{1}{x} \,\mathrm{d}x = \ln(x) + C\)

\(\displaystyle \int \mathrm{e}^x \,\mathrm{d}x = \mathrm{e}^x + C\)

\(\displaystyle \int b^x \,\mathrm{d}x = \tfrac{1}{\ln(b)}b^x + C\)

\(\displaystyle \int \cos(x) \,\mathrm{d}x = \sin(x) + C\)

\(\displaystyle \int \sin(x) \,\mathrm{d}x = -\cos(x) + C\)

Like differentiation, antidifferentiation is a linear operation: constants factor out and it splits across addition/subtraction.

\(\displaystyle \int kf \,\mathrm{d}x = k \int f \,\mathrm{d}x \)

\(\displaystyle \int \bigl(f \pm g\bigr) \,\mathrm{d}x = \int f \,\mathrm{d}x \pm \int g \,\mathrm{d}x \)