me22b113.tex

\section*{ME22B113}
Name: Bhavesh Rachaputi \\
GitHub ID: Bhavesh1056 \\
\subsection*{\textbf{Leibniz Integral Rule}}
\begin{equation}
\frac{\mathrm{d} }{\mathrm{d} x}\left ( \int_{a(x)}^{b(x)}f(x,t) dt \right) = f(x,b(x)) \cdot \frac{\mathrm{d} }{\mathrm{d} x} b(x) - f(x,a(x)) \cdot \frac{\mathrm{d} }{\mathrm{d} x} a(x) + \int_{a(x)}^{b(x)} \frac{\partial}{\partial x} f(x,t) dt
\end{equation}
\footnote{Weisstein, E. W. (2003). Leibniz integral rule.
https://mathworld.wolfram.com/LeibnizIntegralRule.html}This is the Leibniz Intergal rule for differentiation under the integral sign where, $-\infty < a(x),b(x) < \infty$ and the integrands are functions dependent on x.\\ In the special case where the functions a(x) and b(x) are constants a(x)=a and b(x)=b with values that do not depend on x, the right hand side of the equation simplifies to include only the last term.\\ \footnote{https://en.wikipedia.org/wiki/Leibniz\_integral\_rule}This important result may, under certain conditions, be used to interchange the integral and partial differential operators, and is particularly useful in the differentiation of integral transforms.