$\"\"$

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The Laplace tranformation of $\"\"$ is $\"\"$ .

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$\"\"$ is a function continuous for $\"\"$ .

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(a) Find the Laplace transforms of $\"\"$ .

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The Laplace transformation of $\"\"$ is $\"\"$ .

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Substitute $\"\"$ .

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$\"\"$

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$\"\"$

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The improper integral $\"\"$ is called convergent if the corresponding limit exists.

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$\"\"$ converges only when $\"\"$ .

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The domain of the function is $\"\"$ .

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When $\"\"$ ,

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$\"\"$

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$\"\"$

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$\"\"$ .

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The Laplace transforms of $\"\"$ is $\"\"$ .

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The domain of $\"\"$ is $\"\"$ .

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$\"\"$

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(b) Find the Laplace transforms of $\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ converges only when $\"\"$ .

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The domain of the function is $\"\"$ .

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When $\"\"$ ,

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$\"\"$

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$\"\"$ .

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The Laplace transforms of $\"\"$ is $\"\"$ .

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$\"\"$

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(c) Find the Laplace transforms of $\"\"$ .

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Substitute $\"\"$ in $\"\"$ .

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$\"\"$

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$\"\"$ .

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Solve the integral by using parts of integration method.

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Formula for integration by parts : $\"\"$ .

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$\"\"$ and $\"\"$ .

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Consider $\"\"$ .

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Apply derivative on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$ .

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Consider $\"\"$ .

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Apply integral on each side.

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$\"\"$

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$\"\"$ .

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Substitute the corresponding values in $\"\"$ .

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ converges only when $\"\"$ .

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The domain of the function is $\"\"$ .

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When $\"\"$ ,

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$ .

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The Laplace transforms of $\"\"$ is $\"\"$ .

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$\"\"$

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(a) $\"\"$ ; Domain $\"\"$ .

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(b) $\"\"$ ; Domain $\"\"$ .

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(c) $\"\"$ ; Domain $\"\"$ .