Question 4.10: One-Dimensional Leibniz Integration Reduce the following exp...

F(t) is to be evaluated from the given expression.

Analysis The integral is

F(t)=\frac{d}{d t} \int_{x=A t}^{x=B t} e^{-2 x^{2}} d x (1)

We could try integrating first, and then differentiating, but we can instead use the 1-D Leibniz theorem. Here, G(x, t)=e^{-2 x^{2}}(G is not a function of time in this simple example). The limits of integration are a(t) = At and b(t) = Bt. Thus,

\begin{aligned}F(t) &=\int_{a}^{b} \frac{\partial G}{\partial t} d x+\frac{d b}{d t} G(b, t)-\frac{d a}{d t} G(a, t) \\&=0 \quad+B e^{-2 b^{2}}-A e^{-2 a^{2}}\end{aligned} (2)

or

F(t)=B e^{-2 B^{2} t^{2}}-A e^{-2 A^{2} t^{2}} (3)

Discussion You are welcome to try to obtain the same solution without using the Leibniz theorem.