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The solution determines the Fourier coefficients for a periodic square pulse with a width of 3 and period of 12, finding the values of α, β, and λ in the given formula for ak.

The derivative of tan(θ) with respect to θ is sec^2(θ), derived using right-angled triangles and the definition of the derivative.

The provided examples demonstrate the continuity of functions at specific points. Polynomials are continuous everywhere, functions with undefined points are not continuous at those points, and absolute value functions are continuous everywhere, including at points where the expression inside the absolute value is zero.

The double integral, using polar coordinates and integration by parts, evaluates to π(e^2 + 1).

Roulette outcomes are random and independent, making future predictions impossible based on past results. Each number has an equal probability of appearing.

The solution to the system of linear equations is x = 1, y = -2, z = 3, and w = 2.

Given a^2 - 4 = 0, the possible values for a are 2 and -2. Substituting these values into a^3 - 4, we find that a^3 - 4 can be either 4 or -12.

The solution attempts to derive a reduction formula for the integral \(\int_0^\infty \frac{x}{\cosh^n(x)} dx\) using integration by parts and Feynman's trick, but ultimately finds that expressing the integral in terms of hypergeometric functions is a more suitable approach.

The integral \int_{0}^{\infty} \frac{x}{(\cosh x)^2} dx evaluates to ln(2).

To evaluate the limit, direct substitution results in an indeterminate form (0/0). Factoring the numerator reveals a common factor of (x-3), which cancels with the denominator. Substituting x=3 into the simplified expression yields a final answer of 58.