Evaluating Complex Integrals Using Partial Fractions and Integration Techniques

Integral calculus is a fundamental tool in advanced mathematics, particularly in engineering, physics, and other scientific fields. This article delves into the evaluation of a specific complex integral, demonstrating the application of partial fractions and integration techniques. The focus is on understanding and solving integrals of the form:

Integral Representation and Simplification

The given integral is represented as:

[ I int_{-infty}^{infty} frac{x^2}{1x^2^2x^22x2} , dx ]

For simplicity, we can split this integral into two parts, focusing on the symmetry and properties of the integrand:

[ I int_{0}^{infty} frac{x^2}{1x^2^2x^44} , dx ]

To simplify the integrand, let us introduce a substitution. Let

[ x^2 y ]

Then, the integral transforms into:

[ I int_{0}^{infty} frac{2y}{1y^24} , dy ]

We further break down the integrand into partial fractions:

[ frac{2y}{1y^24} frac{A}{1y^2} frac{B}{1y} frac{2Cy^24D}{y^24} ]

Evaluation of the Integral

We start with the partial fraction decomposition and focus on the evaluation of the integrals separately. We shall not find the constants initially. Due to the peculiar nature of the problem, the constants C and D are not necessary for evaluation; only the combination of these constants is needed. We separate the integral into three parts for clarity:

[ I int_{0}^{infty} left( color{red} frac{A}{1y^2} color{cyan} frac{B}{1y} color{green} frac{2Cy^44D}{y^24} right) , dy ]

First, consider the integral:

[ color{red} int_{0}^{infty} frac{1}{1y^2} , dy ]

Using the known result for the integral of this form:

[ int_{0}^{infty} frac{1}{1y^2} , dy frac{pi}{4} ]

Next, evaluate:

[ color{cyan} int_{0}^{infty} frac{1}{1y} , dy tan^{-1}(y) bigg|_{0}^{infty} frac{pi}{2} ]

And finally, the integral:

[ color{green} int_{0}^{infty} frac{2Cy^44D}{y^24} , dy ]

Substitute

[ frac{2Cy^44D}{y^24} frac{CD}{2} left( frac{1}{y^2-4} right) ]

Which simplifies to:

[ color{green} int_{0}^{infty} frac{2Cy^44D}{y^24} , dy frac{CD}{2} pi ]

By combining these results, we determine the constants:

[ 2y^2 y^24 left[ A Bfrac{1}{y} right] ]

Plug in ( y -1 ) to find:

[ -2 5A quad Rightarrow quad A -frac{2}{5} ]

Differentiate with respect to ( y ) to find:

[ 4y^2 2A B y ]

Plug in ( y -1 ) to find:

[ 0 -2A - B quad Rightarrow quad B -frac{4}{25} ]

Plug in ( y 2 ) to find:

[ C frac{32}{25} ]

A detailed integration yields:

[ I color{red} frac{Api}{4} color{cyan} frac{Bpi}{2} color{green} frac{CD}{2} pi frac{7pi}{50} ]

This step-by-step procedure illustrates the method for evaluating complex integrals using partial fractions and integration techniques. The result shows that the integral evaluates to ( frac{7pi}{50} )).