1. The Laplace transform F(s) of the exponential function. f(t) = eat when t ≥ 0, where a is a constant and (s - a) > 0, is
2. Evaluate $$\int\limits_0^\infty {\frac{{\sin {\text{t}}}}{{\text{t}}}{\text{dt}}} $$
3. If the Laplace transform of $${{\text{e}}^{\omega {\text{t}}}}$$ is $$\frac{1}{{{\text{s}} - \omega }},$$ the Laplace transform of tcosh t is
4. 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
5. The Laplace transform of ei5t where $${\text{i}} = \sqrt { - 1} ,$$ is
6. Laplace transform of the function sin ωt is
7. For the function \[{\text{f}}\left( {\text{x}} \right) = \left\{ {\begin{array}{*{20}{c}}
{ - 2,}&{ - \pi < {\text{x}} < 0} \\
{2,}&{0 < {\text{x}} < \pi }
\end{array}} \right.\]
The value of an in the Fourier series expansion of f(x) is
The value of an in the Fourier series expansion of f(x) is
8. Laplace transform of cos (ωt) is $$\frac{{\text{s}}}{{{{\text{s}}^2} + {\omega ^2}}}.$$ The Laplace transform of e-2t cos(4t) is
9. The Laplace transform of sinh(at) is
10. 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 value of the coefficient a2 for the function f(x) = cos2(x) in [0, π] is