Table 4.4.1 lists the rule for making a substitution, or change of variable, in an integral.

If $F\left(y\right)$ is the antiderivative obtained when the substitution $x\=g\left(y\right)$ is made in the indefinite integral, then $F\left(g^{-1}\left(x\right)\right)$ is an indefinite integral in terms of the original variable, $x$; correspondingly, the value of the definite integral is given by $F\left(g^{-1}\left(b\right)\right)-F\left(g^{-1}\left(a\right)\right)$ = $F\left(g^{-1}\left(x\right)\right){\|}_a^b$.

Alternate forms for the substitution are $y\=h\left(x\right)$ and $r\left(y\right)\=s\left(x\right)$. In either of these cases, obtain $\mathrm{dx}$ by solving explicitly for $x$ and differentiating, or by differentiating implicitly. No matter how the substitution rule is given, an explicit solution for $y\=y\left(x\right)$ is needed for expressing the indefinite integral in terms of $x$.

Table 4.4.2 details the value of a definite integral of an odd or even function taken over a symmetric interval such as $\left[-a\,a\right]$.

In each case, the region shaded in yellow to the left of the $y$-axis represents the value of the definite integral on $\left[-a\,0\right]$; the region shaded in red to the right of the $y$-axis, the value of the definite integral on $\left[0\,a\right]$. For the even function, the yellow and red regions match exactly, but for the odd function, the yellow and red regions match in magnitude, but differ in sign.