Thinking Backwards Examples

Find  given that

Pull out the constant from the integral to get

We're trying to 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:

Find  given that

Split up the integral:

We know , so

Combine the tricks used in the previous two examples to find

given that

Split up the integral and pull out the constant 5 from the second term:

We know

so we can put that in to get

Since , it's time to solve:

Find  assuming that f is even and that

First switch the limits and throw a minus sign out front:

Since this needs to be 16, and f is even,  must be -8.