This page defines the indefinite integral. Unlike its definite counterpart that calculates the area under a curve, an indefinite integral finds a general expression for the area under the curve. It is therefore without the bounds of the definite integral, as can be seen in the equation below.
| \( x \) | This is a symbol for any generic variable. It can hold any value, whether that be an integer or a real number, or a complex number, or a matrix etc. |
| \( i \) | This is the symbol for an iterator, a variable that changes value to refer to a sequence of elements. |
| \( \int \) | This is the symbol for an integral, sometimes referred to as an antiderivative. Graphically, it can be understood as the area between a curve and the axis the integral is taken with respect to. |
| \( C \) | This symbol represents the constant of integration. It must be added to the result of all definite integrals to encompass all possible solutions that satisfy the integral. |
| \( \sum \) | This is the summation symbol in mathematics, it represents the sum of a sequence of numbers. |
| \( f \) | This is the symbol for a function. It is commonly used in algebra, and (multivariate) calculus. |
| \( \lim \) | This is the symbol for a limit in calculus. It's a function that models an output, as an input approaches a certain value. |
| \( \:d \) | This is the symbol for a differential. It represents an infinitesimally small (infinitely close to zero) change in whatever variable it is with respect to. |
| \( n \) | This is the symbol for the number of subintervals an area is broken up into. |
| \( \Delta \) | This is the symbol for the amount that some variable changes. |
| \( \infty \) | This is the symbol for infinity, a concept representing the idea of something without bound or end. It represents an unbounded quantity larger than any real number. |
Consider the indefinite integral:
\[\htmlClass{sdt-0000000060}{\int}_{\htmlClass{sdt-0000000121}{a}}^{\htmlClass{sdt-0000000033}{b}} \htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000003}{x}) \htmlClass{sdt-0000000102}{\:d} \htmlClass{sdt-0000000003}{x} = \htmlClass{sdt-0000000101}{\lim}_{\htmlClass{sdt-0000000104}{n} \to \htmlClass{sdt-0000000108}{\infty}} \htmlClass{sdt-0000000080}{\sum}_{\htmlClass{sdt-0000000018}{i} = 1}^{\htmlClass{sdt-0000000104}{n}} \htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000003}{x}_{\htmlClass{sdt-0000000018}{i}}) \htmlClass{sdt-0000000105}{\Delta} \htmlClass{sdt-0000000003}{x}_{\htmlClass{sdt-0000000018}{i}}\]
Here, \( \htmlClass{sdt-0000000104}{n} \) refers to the number of intervals between \( \htmlClass{sdt-0000000121}{a} \)t and \( \htmlClass{sdt-0000000033}{b} \). However, we are not interested in \( \htmlClass{sdt-0000000121}{a} \) and \( \htmlClass{sdt-0000000033}{b} \) as we want a general expression, so instead we just ignore them, and analytically continue with the rest of the expression. We therefore have the same equation, just without the limits on the integral:
\[ \htmlClass{sdt-0000000060}{\int} \htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000003}{x}) \htmlClass{sdt-0000000102}{\:d} \htmlClass{sdt-0000000003}{x} = \htmlClass{sdt-0000000101}{\lim}_{\htmlClass{sdt-0000000104}{n} \to \htmlClass{sdt-0000000108}{\infty}}\htmlClass{sdt-0000000080}{\sum}_{\htmlClass{sdt-0000000018}{i} = 1}^{\htmlClass{sdt-0000000104}{n}} \htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000003}{x}_{\htmlClass{sdt-0000000018}{i}}) \htmlClass{sdt-0000000105}{\Delta} \htmlClass{sdt-0000000003}{x}_{\htmlClass{sdt-0000000018}{i}} \]
However, because there are now an infinite number of solutions to the integral, just by adding a constant, we have to say that the integral is:
\[ \htmlClass{sdt-0000000060}{\int} \htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000003}{x}) \htmlClass{sdt-0000000102}{\:d} \htmlClass{sdt-0000000003}{x} = (\htmlClass{sdt-0000000101}{\lim}_{\htmlClass{sdt-0000000104}{n} \to \htmlClass{sdt-0000000108}{\infty}}\htmlClass{sdt-0000000080}{\sum}_{\htmlClass{sdt-0000000018}{i} = 1}^{\htmlClass{sdt-0000000104}{n}} \htmlClass{sdt-0000000096}{f}(\htmlClass{sdt-0000000003}{x}_{\htmlClass{sdt-0000000018}{i}}) \htmlClass{sdt-0000000105}{\Delta} \htmlClass{sdt-0000000003}{x}_{\htmlClass{sdt-0000000018}{i}}) + \htmlClass{sdt-0000000070}{C} \]
as required.
Coming soon...