Question

Derivatives of integrals Simplify the following expressions. $$\frac{d}{d x} \int_{x}^{0} \frac{d p}{p^{2}+1}$$

Short answer

Answer: The derivative is \(-\frac{d x}{x^{2}+1}\).

Step-by-step solution

Recognize the Fundamental Theorem of Calculus (Part 1)

Since we are asked to find the derivative of an integral function, we can use the Fundamental Theorem of Calculus Part 1: $$\frac{d}{d x} \int_{a}^{x} f(t) dt = f(x)$$ In our case, the expression is: $$\frac{d}{d x} \int_{x}^{0} \frac{d p}{p^{2}+1}$$

Rewrite the integral with the correct lower and upper limits

To use the Fundamental Theorem of Calculus Part 1, we need to rewrite our integral with the lower and upper limits in the standard order (lower limit before upper limit). We do this by switching the limits and adding a negative sign: $$\frac{d}{d x} \left(-\int_{0}^{x} \frac{d p}{p^{2}+1}\right)$$ Now we can apply the theorem:

Apply the Fundamental Theorem of Calculus Part 1

We have: $$\frac{d}{d x} \left(-\int_{0}^{x} \frac{d p}{p^{2}+1}\right)$$ According to the theorem: $$\frac{d}{d x} \left(-\int_{0}^{x} \frac{d p}{p^{2}+1}\right) = -\frac{d x}{x^{2}+1}$$ So, the simplified expression is: $$-\frac{d x}{x^{2}+1}$$

Key concepts

Derivative of an Integral

Understanding the relationship between a derivative and an integral is a cornerstone of calculus. The Fundamental Theorem of Calculus bridges this connection beautifully, where the first part states that the derivative of an integral whose upper limit is a variable is simply the original function. In essence, if we have an integral of the form
\[ \frac{d}{dx} \int_{a}^{x} f(t) dt \]
the derivative of this integral with respect to x is simply
\[ f(x) \].
However, if the lower limit is the variable, as in our exercise, we must rewrite the integral by reversing the limits and introducing a negative sign, leading us to
\[ \frac{d}{dx} \left(-\int_{0}^{x} \frac{dp}{p^{2}+1}\right) \]
Thus, applying the theorem gives us the derived function, which is the integrand evaluated at the upper limit.

Integration Limits

The integration limits define the boundaries over which the function is integrated. When dealing with definite integrals, the lower limit is typically listed first followed by the upper limit. In the context of the exercise, there's a notable distinction: the integration starts at the variable and moves toward a constant. Mathematically, this requires us to change the order to use the Fundamental Theorem of Calculus effectively. Reversing the limits from
\[ \int_{x}^{0} \frac{dp}{p^{2}+1} \]
to
\[ \int_{0}^{x} \frac{dp}{p^{2}+1} \]
introduces a minus sign, acknowledging that the direction of integration has been flipped. Failing to account for this reversal would result in an incorrect application of the theorem and therefore, an incorrect solution.

Integrating Functions

The process of integration entails finding the antiderivative or the area under the curve of a given function. It is the inverse operation of taking a derivative. When integrating functions with respect to x, it is important to consider the function within the context of its variables. For instance, in our exercise, we integrate the function
\[ \frac{1}{p^{2}+1} \]
with respect to p. This particular function, when integrated, yields
\[ \tan^{-1}(p) \],
but since we are interested in taking the derivative with respect to x, it simplifies directly via the Fundamental Theorem of Calculus without needing to find this antiderivative explicitly.

Calculus Theorems

Calculus is underpinned by several key theorems that govern the behavior of functions, derivatives, and integrals. In particular, the Fundamental Theorem of Calculus serves as the backbone, connecting differentiation and integration in a single elegant framework. It comprises two parts: the first deals with taking derivatives of integrals as demonstrated in our exercise, while the second part informs us how to evaluate definite integrals using antiderivatives. Other important theorems include the Mean Value Theorem, which, under certain conditions, guarantees the existence of a point where a function's instantaneous rate of change equals its average rate of change over an interval, and Taylor's Theorem, which gives us a powerful tool to approximate functions using polynomials.