This article presents a detailed analysis of two advanced problems located at the intersection of mathematical analysis and linear algebra. The first problem,demonstrates the integration of a logarithmic function with a rational expression, solved using series expansion techniques. The second problem involves the integration of a matrix-valued cosine function under the Gaussian kernel, offering an approach to operator-valued functions and matrix analysis. Both problems are designed to enhance students’ theoretical knowledge, logical reasoning, and familiarity with competition-level mathematical problem-solving. The article serves as a methodological guide for gifted learners aiming to deepen their understanding of advanced mathematical concepts.

Mathematical analysis, definite integral, logarithmic function, matrix-valued function, linear algebra, operator functions, olympiad problems, methodological approach, series expansion, analytical solution.

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