1. Sometimes we work with tricky integrals. For example, consider the integral
What is ???
We can evaluate (1) directly as
Thus its derivative is
But what about the general case?
2. The general scheme is quite simple. We write
Then it follows that
How can we see this?
Let be the antiderivative for , so . Then we see (4) becomes
Take the derivative of both sides (using the chain rule on the right hand side) gives us
precisely as desired!
Example 1 [Math.SE]. Consider the function
What is its derivative?
Solution. We see that
This is probably done too quick, but we only need to differentiate the “boundary functions” and . These are trivial calculations.
Remark. Observe by the fundamental theorem of calculus we have
which is unexciting, until you substitute Equation (9) in and obtain
So what? Well, we should see that , so we have another definition for , namely
We found a way to relate two integrals together!
3. Usefulness. Usually we have to calculate a hard integral but cannot do it directly. So it helps to differentiate under the integral sign, then integrate the result using the fundamental theorem of calculus. This trick is sometimes called “Integration under the Integral Sign” and solves a lot of problems!
Example 2. Consider the integral
We can integrate over which gives us
Switching the order of integration gives us
Thus we deduce, setting (13) equal to (14), that
This is, quite literally, the textbook example (see Woods’ Advanced Calculus text, section 61).