"Thinking backwards" shows up in a lot of places. That means it must be important! Here, "thinking backwards" means instead of using a known integral value to evaluate an expression, we'll work backwards from an equation to find the integral value.
Think of the integral you're looking for as a unit. Instead of solving for x, you'll be solving for something like 
. If your equation contains a slightly different integral than the one you want, use the properties to manipulate the integral until it's what you're looking for.
Example 1
Find
|
, which means we need to get that all by itself on one side of the equation. Divide both sides by 2, and there we are:
Example 2
Find
|
given that
, so
Example 3
Combine the tricks used in the previous two examples to find
given that
|
, it's time to solve:
Example 4
Find
|
assuming that f is even and that
must be -8.
given that
given that
given that
by looking at a graph. The region between x and the x-axis on [0, 3] is a triangle with base 3 and height 3, so its area is 4.5.
given that f (x) is even and that
.
given that f is odd and
