if $a&ne#neq;-1$

Just as with the power rule, many other rules and results that you
already know yield techniques for integration. In general
integration is potentially much trickier than differentiation,
because it is often not obvious which technique to use, or even
how to use it. *Integration is a more exciting than differentiation!*

Recall the *chain
rule*, which asserts that

If then
, and the substitution rule simply
says if you let formally in the integral everywhere, what you
naturally would hope to be true based on the notation actually is
true. The substitution rule illustrates how the notation Leibniz
invented for Calculus is *incredibly brilliant*. It is said that
Leibniz would often spend days just trying to find the right notation
for a concept. He succeeded.

As with all of Calculus, the best way to start to get your head around
a new concept is to see severally clearly worked out examples. (And
the best way to actually be able to use the new idea is to *do*
lots of problems yourself!) In this section we present examples that
illustrate how to apply the substituion rule to compute indefinite
integrals.

William Stein 2006-03-15