Theorem (direct substitution)
Let f be a function defined on an interval I which has an antiderivative F there. Let g be a function from an interval J into the interval I which is differentiable on J. Then F(g) is an antiderivative of f (g)g′ on J:

Note: This theorem tells us what happens if we try to transform the variable (that is, pass from one variable to another) using the transformation y = g(x). It turns out that the differential dy of the new variable is given by the equation dy = g′(x)dx. It actually seeems natural, since if we differentiate both sides of the equation y = g(x) using the Leibniz notation, we get a (symbolic!) equality