Evaluate ∫₀¹ X Arctan(x) Li₂((1-x²)/2) / (1+x²) Dx

Let's embark on a journey to evaluate the intricate definite integral: ∫₀¹ (x arctan(x) Li₂( (1-x²)/2 ) ) / (1+x²) dx. This problem combines elements of real analysis, calculus, integration techniques, and polylogarithm functions, making it a fascinating challenge. We aim to understand the steps and reasoning required to arrive at the solution: -1/2 β(4) + (π²/16)G + (3π/128) - ..., where β(4) is the Dirichlet beta function and G is Catalan's constant.

Understanding the Integral's Components

Before diving into the solution, let's break down the integral's components:

• x arctan(x) / (1+x²): This part involves a rational function multiplied by the arctangent function. The presence of arctan(x) suggests that trigonometric substitution or integration by parts involving derivatives and integrals of arctan(x) might be useful.
• Li₂((1-x²)/2): This is the dilogarithm function, denoted as Li₂(z), which is defined as Li₂(z) = -∫₀ᶻ ln(1-t)/t dt. Understanding the properties and potential series representation of the dilogarithm is crucial.
• The limits of integration, 0 to 1: These limits define the interval over which we are evaluating the integral. They often play a key role in simplifying the result or suggesting specific integration techniques.

Potential Strategies and Techniques

To tackle this integral, we can consider the following strategies:

1. Substitution: Look for suitable substitutions that simplify the integrand. Given the presence of (1-x²)/2, a trigonometric substitution like x = sin(θ) or x = cos(θ) might be helpful. Also, substituting u = arctan(x) could simplify the arctan(x) term.
2. Integration by Parts: Applying integration by parts could help shift derivatives from one part of the integrand to another, potentially simplifying the expression. The choice of u and dv is critical in this method.
3. Series Expansion: Expanding the dilogarithm function Li₂((1-x²)/2) into its series representation might allow us to integrate term by term. Recall that Li₂(z) = ∑ₖ<binary data, 1 bytes><binary data, 1 bytes><binary data, 1 bytes>₁^∞ zᵏ / k².
4. Special Functions and Identities: Utilize known properties and identities of the dilogarithm function and other special functions to simplify the integral. For instance, there are various identities involving Li₂(z) that relate it to other functions or different arguments.
5. Contour Integration (Advanced): While less likely for a real definite integral of this form, contour integration techniques might provide an alternative approach, especially if the integral can be related to a complex function with known singularities.

Detailed Exploration of Solution Approaches

Let's explore some of these strategies in more detail.

Trigonometric Substitution

Consider the substitution x = sin(θ). Then, dx = cos(θ) dθ, and the limits of integration change from 0 to 1 for x to 0 to π/2 for θ. The integral becomes:

∫₀^(π/2) (sin(θ) arctan(sin(θ)) Li₂((1-sin²(θ))/2) cos(θ)) / (1+sin²(θ)) dθ

Simplifying, we get:

∫₀^(π/2) (sin(θ) cos(θ) arctan(sin(θ)) Li₂(cos²(θ)/2)) / (1+sin²(θ)) dθ

This form might be easier to work with, especially if we can simplify Li₂(cos²(θ)/2) using known identities. One such identity is cos²(θ) = (1 + cos(2θ))/2, which could lead to further simplification.

Integration by Parts

Applying integration by parts requires choosing appropriate functions for u and dv. Let's try:

• u = Li₂((1-x²)/2)
• dv = (x arctan(x)) / (1+x²) dx

Then, we need to find du and v. First, let's find du:

du/dx = d/dx [Li₂((1-x²)/2)] = -ln(1 - (1-x²)/2) / ((1-x²)/2) * (-2x/2) = (2x ln((1+x²)/2)) / (1-x²)

Now, let's find v:

v = ∫ (x arctan(x)) / (1+x²) dx

To evaluate this integral, let w = arctan(x), so x = tan(w) and dx = sec²(w) dw = (1+tan²(w)) dw = (1+x²) dw. The integral becomes:

v = ∫ tan(w) * w * (1+tan²(w)) / (1+tan²(w)) dw = ∫ w tan(w) dw

Integrating ∫ w tan(w) dw is not straightforward and might involve special functions. However, let's assume we can find v and proceed with integration by parts:

∫₀¹ u dv = uv |₀¹ - ∫₀¹ v du

This approach might lead to a more manageable integral, especially if v has a simpler form.

Series Expansion of the Dilogarithm

Using the series expansion of the dilogarithm, we have:

Li₂((1-x²)/2) = ∑ₖ<binary data, 1 bytes><binary data, 1 bytes><binary data, 1 bytes>₁^∞ ((1-x²)/2)ᵏ / k² = ∑ₖ<binary data, 1 bytes><binary data, 1 bytes><binary data, 1 bytes>₁^∞ (1-x²)ᵏ / (2ᵏ k²)

Substituting this into the original integral, we get:

∫₀¹ (x arctan(x)) / (1+x²) * ∑ₖ<binary data, 1 bytes><binary data, 1 bytes><binary data, 1 bytes>₁^∞ (1-x²)ᵏ / (2ᵏ k²) dx = ∑ₖ<binary data, 1 bytes><binary data, 1 bytes><binary data, 1 bytes>₁^∞ (1 / (2ᵏ k²)) ∫₀¹ (x arctan(x) (1-x²)ᵏ) / (1+x²) dx

The integral inside the summation might be easier to evaluate for specific values of k. We could then try to find a pattern or a closed-form expression for the sum.

Potential Challenges and Considerations

• Convergence: Ensure that any series expansions used are convergent within the interval of integration.
• Complexity: The integral involves special functions and might require advanced techniques to evaluate fully.
• Computational Tools: Utilizing computer algebra systems (CAS) like Mathematica or Maple can help verify intermediate steps and provide numerical approximations.

The Path to the Solution

While the exact steps to arrive at the solution -1/2 β(4) + (π²/16)G + (3π/128) - ... are complex and involve a combination of these techniques, the process typically includes:

1. Strategic Substitution: To simplify the integrand.
2. Integration by Parts: To shift derivatives and simplify the expression.
3. Series Manipulation: To express the dilogarithm function as a series and potentially integrate term by term.
4. Recognition of Special Functions: Such as the Dirichlet beta function β(4) and Catalan's constant G.

This exploration provides a comprehensive overview of the approaches and techniques required to evaluate the given definite integral. The final solution involves a combination of analytical skills and potentially the use of computational tools to handle the complexity of the expressions.

By systematically applying these strategies, one can navigate the intricate path toward the solution, gaining a deeper appreciation for the interplay between different areas of mathematics.