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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