The Fourier cosine series for an even function f(x) is given by $${\text{f}}\left( {\text{x}} \right) = {{\text{a}}_0} + \sum\limits_{{\text{n}} = 1}^\infty {{{\text{a}}_{\text{n}}}\cos \left( {{\text{nx}}} \right)} $$
The value of the coefficient a₂ for the function f(x) = cos²(x) in [0, π] is

The Laplace transform F(s) of the exponential function. f(t) = e^at when t ≥ 0, where a is a constant and (s - a) > 0, is

Evaluate $$\int\limits_0^\infty {\frac{{\sin {\text{t}}}}{{\text{t}}}{\text{dt}}} $$

If the Laplace transform of $${{\text{e}}^{\omega {\text{t}}}}$$  is $$\frac{1}{{{\text{s}} - \omega }},$$  the Laplace transform of tcosh t is

The function f(t) satisfies the differential equation $$\frac{{{{\text{d}}^2}{\text{f}}}}{{{\text{d}}{{\text{t}}^2}}} + {\text{f}} = 0$$   and the auxiliary conditions, f(0) = 0, $$\frac{{{\text{df}}}}{{{\text{dt}}}}\left( 0 \right) = 4.$$  The Laplace transform of f(t) is given by