The integral of a function, f(x), is symbolized by ∫f(x)dx. The differential dx indicates the variable of integral is always written to the far right.

The expression between the integral sign and the differential is the expression to be integrated. This representation is for a function of a single variable, usually studied in Calc I & II.

In multivariable calculus (calc III), the function to be integrated is usually dependent on several variables, i.e., f(x,y) or f(x,y,z) and multiple integrals appear. For example, ∫∫f(x,y)dxdy and ∫∫∫f(x,y,z)dxdydz represent a double and triple integral in calculus III and can be done by two and three separate integrations, respectively. They are done by separate integrations starting from the inside out. For example, in the case of the triple integral, the first integration is done with respect to x, then with respect to y, and finally with respect to z. The other variables in the expression f(x,y,z) are treated as constants when performing the integration with respect to one of them. For multiple integrals, the order of integrations is always done from the inner most variable out to the variable on the extreme right. (These variables are indicated by their differentials).

These are examples of Indefinite Integrals, since no limits appear on the integration symbol. So, the results will be variable expressions with unknown constants (assuming no conditions on the problem are given). Definite Integrals have limits on the integration symbol (lower & higher). See my link on the definite integral and what it represents.

So, going from a derivative to its parent function will involve integrating the derivative. However, for indefinite integrals, the result will contain many answers since an unknown constant will be present. In symbols, ∫f(x)dx=F(x) + C. I use F(x) in the result since that function will differ from f(x). The constant C is arbitrary and could take on an infinite number of values, if no other information is given in the problem.

It will always be the case that F'(x) = f(x). This is a fundamental relationship that ties both processes together. This is one of the fundamental theorems of calculus. For example, ∫2xdx=x² + C. Note that the derivative of x² is 2x, which is the expression in the integral.

However, only the simplest of integrals can be done by guess work. Rules are established for integration so more complicated integrals can be evaluated. There are expressions that are very complicated & special techniques are studied (calc II) and others that can’t be done. However, most applications deal with definite integrals. Their values can be approximated very closely. The calculator will do this for us very quickly (see link on using the TI-83 in calculus).