Differentiation under integral sign
Motivating example Evaluating following integral $$ I = \int_0^1{\frac{1 - x^2}{\ln{x}}dx} $$ Closed-form results $$ \begin{equation} \begin{aligned} F(t) &= \int_0^1{\frac{1-x^t}{\ln(x)}dx} \\ \implies \frac{d}{dt}F &= \frac{d}{dt}\int_0^1{\frac{1-x^t}{\ln(x)}dx}\\ &= \int_0^1{ \frac{\partial}{\partial t} \frac{1-x^t}{\ln(x)}dx }\\ &= \int_0^1{ \frac{-\ln(x)x^t}{ln(x)} dx} \\ &= \bigg[-\frac{x^{t+1}}{t+1}\bigg]_0^1\\ &= -\frac{1}{t+1}\\ \implies F(t) &= -\ln({t+1}) \\ \implies I &= f(2) = -\ln3 \end{aligned} \end{equation} $$ Numerical approximation Code to produce the figure 1 2 3 4 5 6 7 8 import numpy as np from matplotlib import pyplot as plt def I(): g = lambda x: (1 - x**2)/np....