Integrals containing one of the expressions on the left in Table 6.3.1 may yield to the companion substitution suggested in the middle column of the table. The substitution, called a trig substitution, is based on the related trig identity stated in the rightmost column of the table.

Figures 6.3.1 - 6.3.3 are useful for expressing the basic trig functions in terms of the variables and parameters appearing in the substitutions listed in Table 6.3.1.

Table 6.3.2 lists eight examples for each case given in Table 6.3.1. Each of the three cases in Table 6.3.1 is represented by one of the functions defined at the top of Table 6.3.2. For each of these representative functions, the examples chosen are common instances of indefinite integrals in which the given functions appear. Notice the pattern of the eight examples in each column of Table 6.3.2: The function and its reciprocal, the function multiplied by $x$, divided by $x$, and divided into $x$, then multiplied by $x^2$, divided by $x^2$, and divided into $x^2$.

Note that while $g\left(x\right)$ is real for all real $x$,  $f\left(x\right)$ is real for $\left|x\right|\le 2\/3$, and $h\left(x\right)$ is real for $\left|x\right|\ge 2\/3$.