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At a Glance - Thinking Backwards

"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  given that


Example 2

Find  given that


Example 3

Combine the tricks used in the previous two examples to find

given that


Example 4

Find  assuming that f is even and that


Exercise 1

Find  given that


Exercise 2

Find  given that


Exercise 3

Find  given that


Exercise 4

Find  given that f (x) is even and that


Exercise 5

Find  given that f is odd and


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