Differentiation under the integral sign is an operation in calculus used to evaluate certain integrals. Under fairly loose conditions on the function being integrated, differentiation under the integral sign allows one to interchange the order of integration and differentiation. In its simplest form, called the Leibniz integral rule, differentiation under the integral sign makes the following equation valid under light assumptions on :
Many integrals that would otherwise be impossible or require significantly more complex methods can be solved by this approach.
The most general form of differentiation under the integral sign states that: if is a continuous and continuously differentiable (i.e., partial derivatives exist and are themselves continuous) function and the limits of integration and are continuous and continuously differentiable functions of , then
In the case where and are constant functions, this formula reduces to the simpler form This simpler statement is known as Leibniz integral rule.
Generally, one uses differentiation under the integral sign to evaluate integrals that can be thought of as belonging to some family of integrals parameterized by a real variable. To better understand this statement, consider the following example:
Compute the definite integral
This integral appears resistant to standard integration techniques such as integration by parts, u-substitution, etc. We would like to use differentiation under the integral sign to compute it.
How can we choose a function to differentiate under the integral sign? The appearance of in the denominator of the integrand is quite unwelcome, and we would like to get rid of it. Thankfully, we know so differentiating the numerator with respect to the exponent seems to be what we'd like to do.
Accordingly, we define a function In this notation, the integral we wish to evaluate is . Observe that the given integral has been recast as member of a family of definite integrals indexed by the variable .
By Leibniz integral rule, we compute It follows that for some constant .
To determine , note that , so . Hence, for all such that the integral exists. In particular, .
In the example, part of the integrand was replaced with a variable and the resultant function was studied using differentiation under the integral sign. This is a good illustration of the problem-solving principle: if stuck on a specific problem, try solving a more general problem.
Another example illustrates the power of this technique in its general form, as one may use it to compute the Gaussian integral.
Compute the definite integral
Define a function Our goal is to compute and then take its square root.
Differentiating with respect to gives
Make the change of variables , so that the integral transforms to Now, the integrand has a closed-form antiderivative with respect to :
Set Then by the above calculation, , so . To determine , take in the equation; since and it follows that .
Finally, taking , we conclude . Thus,
Let be real with . Show that
Let be real with . Define
If we set then we find
So we will focus on determining . Recall that
and so, differentiating under the integral sign with respect to we find
Completing the square in the exponent we have
Thus, noting as , we finally have
We will set so that
One should also note counterexamples, for which this technique does not work. For instance, suppose one attempts to evaluate by making the variable change for some . Then, Differentiating under the integral sign yields which is absurd. The problem is that the function is not continuously differentiable (consider when ), which was required in the assumptions set forth above.