DIFFERENTIATING INTEGRALS OR INTEGRATING DERIVATIVES
The following equation (Leibnitz formula) describes how to differentiate an integral (see page
854, §C.3, of Transport Phenomena):
∫
d
dt
a2 (t )
f ( x, t ) dx =
a1 (t )
∫
a2 (t )
a1 (t )
∂ f
d a2
d a1
dx + f (a2 , t )
− f (a1, t )
∂t
dt
dt
This equation can be reorganized to give a formula for integrating a derivative:
∫
a2 (t )
a1 (t )
∂ f
d
dx =
∂t
dt
∫
a2 (t )
a1 (t )
⎛
d a2
d a1 ⎞
− f ( x, t ) x = a
f ( x, t ) dx − ⎜ f ( x, t ) x =a
⎟
2
1 dt
dt
⎝
⎠
If the integration limits (a1 and a2) are not functions of the variable being differentiated (i.e., t in
the above equation), then:
d
dt
z
a2
f ( x , t ) dx =
a1
z
a2
a1
∂f
dx
∂t
If the variable being differentiated and integrated is the same, then differentiation is the inverse
of integration. That is,
d
dx
z
a2
f ( x , t ) dx =
a1
z
a2
a1
∂f
dx = f
∂x
x = a2
x = a1
= f
x = a2
−f
x = a1