On the Leibnitz Rule for Differentiating Under the Integral Sign (2308.09619v1)

Abstract: This Note revisits the Leibnitz integral calculus method based on differentiation under the integral sign with respect to a parameter either already existing or introduced ad hoc. Through several cases exemplifying the method, it is shown that this approach, applicable to regular and, under certain conditions, to improper integrals as well, results in a 1st order differential equation whose solution is usually straightforward.

• The paper demonstrates that differentiating under the integral sign reduces complex integrals to manageable first-order differential equations.
• It validates the approach with both regular and improper integrals through multiple detailed case studies.
• Results include simplifications leading to known constants, emphasizing the practical utility and versatility of the technique.

On the Leibnitz Rule for Differentiating Under the Integral Sign

This paper revisits the Leibnitz integral calculus method for differentiating under the integral sign, using a parameter either pre-existing or introduced ad hoc. The approach is demonstrated across various case studies, establishing its applicability to both regular and, under certain conditions, improper integrals. The method results in a first-order differential equation, whose solution is typically straightforward. It exemplifies the utility of this technique for simplifying complex integral evaluations.

Summary of Method

Differentiation under the integral sign involves treating the integral as a function of an auxiliary parameter $\alpha$, which can be naturally present or introduced to simplify the problem. The method proceeds by:

1. Parameter Introduction: Consider the original integral as a function of $\alpha$.
2. Evaluation at Specific Values: Evaluate the integral for convenient values of $\alpha$.
3. Validation: Ensure compliance with conditions allowing the interchange of integration and differentiation.
4. Differentiation and Simplification: Differentiate with respect to $\alpha$ and simplify the integral.
5. ODE Formation: Formulate a first-order ordinary differential equation (ODE).
6. Solution Reconstruction: Solve the ODE for any $\alpha$ and apply conditions to remove integration constants.

Four illustrative examples showcase the versatility of this approach.

Practical Utility and Examples

Example 1: Improper Integral Simplification

The method is illustrated with an improper integral:

$$I = \int_{0}^{\infty} \ln(1 + x^{2}) \frac{dx}{x^{2}}$$

By introducing $\alpha$ into the integrand, differentiating yields a simpler integral which leads to the solution $I = \pi$. This demonstrates the straightforward nature in which complex integrals can be reduced to simpler forms through differentiation under the integral sign.

Example 2: Definite Integral with Built-in Parameter

Consider:

$$I(\alpha) = \int_{0}^{\pi} \ln(1 - 2\alpha \cos x + \alpha^2) dx$$

For $|\alpha| \geq 1$, differentiation yields:

$I(\alpha) = 2\pi \ln|\alpha|$

This solution is achieved by re-representing trigonometric terms as complex exponentials and reduces to a first-order ODE, illustrating the method’s effectiveness even in the presence of multi-dimensional integrands and complex numbers.

Example 3: Non-trivial Improper Integral

An ad hoc parameter is introduced:

$$I = \int_{0}^{\infty} \frac{e^{-x^2 \sin(x^2)}}{x^2} dx$$

Following parameter introduction and simplification, the solution involves Gaussian integrals and complex transformations, yielding:

$I = \sqrt{\frac{\pi}{2} \sqrt{\sqrt{2} - 1}}$

Changes in auxiliary parameter placement and integral bounds illustrate alternative routes to solving challenging integrals using this technique.

Example 4: Difficult Definite Integral

The integration is over:

$$I(a) = \int_{-a}^{a} \frac{\ln(1 + x)}{\sqrt{a^2 - x^2}} dx$$

Introducing the parameter $b$ leads to:

$$I(a) = \pi \ln \left( \frac{1 + \sqrt{1 - a^2}}{2} \right)$$

Using trigonometric substitution simplifies the integral and leads to complex logarithmic solutions, effectively solving the problem over defined parameter constraints.

Conclusion

Differentiating under the integral sign is a powerful and efficient method for tackling complex integral calculations. By enforcing the Leibnitz rule and transforming integration problems into differential equations, this method simplifies many integrations. It's a key technique in advanced calculus, providing straightforward solutions to problems that may otherwise involve complicated methodologies. It’s favored for its simplicity and broad applicability across different types and bounds of integrals, both regular and improper.