1. Integration & Logarithmic Functions | Evaluating \\\\(\\\\int (5x - 1)^{-2} dx\\\\) and expressing the result in logarithmic form with integer coefficients.

2. Differentiation & Curve Properties | Computing first and second derivatives of a logarithmic function, verifying slope conditions at a specific point.

3. Arc Length of a Parametric Curve | Finding the exact length of a curve given by parametric equations over a defined interval.

4. Complex Numbers & Polynomial Roots | Using De Moivre’s theorem to derive trigonometric identities and solving a polynomial equation in terms of cotangent.

5. Differential Equations & Initial Conditions | Solving a second-order differential equation with specified initial conditions to determine a particular solution.

6. Eigenvalues & Matrix Factorization | Finding the characteristic equation of a matrix, computing eigenvalues, and expressing matrix powers using diagonalization.