Derivative of an Integral Involving a Rational Function

Find the derivative of the integral from 0 to x of $\frac{T}{1 + T^3}$ with respect to $x$.

This problem deals with a fascinating concept from calculus where we use the Fundamental Theorem of Calculus to find the derivative of an integral. The function being integrated, T over one plus T cubed, is a rational function, which is continuous over the interval and ensures that the integral is well-defined. By taking the derivative of this integral with respect to x, you will essentially apply the Fundamental Theorem of Calculus Part 1, which states that if you have an integral of a function from a constant to x, the derivative of this integral is simply the integrand evaluated at x. This problem highlights the power and elegance of the Fundamental Theorem of Calculus in simplifying the process of differentiation of integrals, effectively linking the operations of integration and differentiation as inverse processes.

In tackling this problem, think of the fundamental relationship between the areas under curves and the slopes of tangent lines—two central ideas in calculus. Conceptually, the differentiation of this integral asks how rapidly the accumulated area under the curve changes as x changes. The answer is straightforward here, thanks to the theorem, but understanding why this relationship holds is crucial to mastering calculus concepts. This problem often reinforces the student's understanding of the integral and derivative as interconnected entities and underlines the utility of rational functions in calculus.