Leibniz Integral Rule
If , then
Theorem 3.8
For a family of functions where is a parameter, define
Then
Proof
By definition,
Split the first integral as an integral over , plus an increment at the upper limit, minus an increment at the lower limit. The difference of integrands over the original interval yields
The upper-limit increment is by the mean-value theorem, so its contribution is . The lower-limit increment contributes .
Examples
For ,
For , the family with has constant limits, so
Setting gives .
Related
Stated in
- Theorem 3.8§3.5 Differentiation of an Integral w.r.t. a Parameter