21. Laplace transform of sin ht is
22. Laplace transform of cos (ωt) is
23. The Laplace transform of a function f(t) is $$\frac{1}{{{{\text{s}}^2}\left( {{\text{s}} + 1} \right)}}.$$ The function f(t) is
24. The inverse Laplace transform of $${\text{H}}\left( {\text{s}} \right) = \frac{{{\text{s}} + 3}}{{{{\text{s}}^2} + 2{\text{s}} + 1}}$$ for t ≥ 0 is
25. Let \[{\text{f}}\left( {\text{x}} \right) = \left\{ {\begin{array}{*{20}{c}}
{ - \pi ,\,\,{\text{if}}}&{ - \pi < {\text{x}} \leqslant {\text{0}}} \\
{\pi ,\,\,{\text{if}}}&{0 < {\text{x}} \leqslant \pi }
\end{array}} \right.\] be a periodic function of period 2π. The coefficient of sin5x in the Fourier series expansion of f(x) in the interval [-π, π] is
26. Given the Fourier series in (-π, π) for f(x) = x cosx, the value of a0 will be
27. Laplace transform of the function f(t) is given by $${\text{F}}\left( {\text{s}} \right) = {\text{L}}\left\{ {{\text{f}}\left( {\text{t}} \right)} \right\} = \int_0^\infty {{\text{f}}\left( {\text{t}} \right){{\text{e}}^{ - {\text{st}}}}{\text{dt}}{\text{.}}} $$ Laplace transform of the function shown below is given by
28. If L defines the Laplace Transform of a function, L [sin (at)] will be equal to
29. The Fourier series expansion of the saw-toothed waveform f(x) = x in (-π, π) of period 2π gives the series, $$1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \,....$$
The sum is equal to
The sum is equal to