$\"\"$

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$\"\"$ for $\"\"$ , where $\"\"$ and $\"\"$ are constants.

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Consider $\"\"$ , such that $\"\"$ .

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If $\"\"$ , then $\"\"$ for all values of $\"\"$ .

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$\"\"$ .

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Multiply on each side by $\"\"$ .

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$\"\"$

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Find the integral $\"\"$ .

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Definition of improper integral type 1:

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$\"\"$ .

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$\"\"$

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$\"\"$

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If $\"\"$ , then $\"\"$ .

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$\"\"$

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Thus, $\"\"$ is a finite number .

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$\"\"$ is convergent by the definition of improper integral of type 1.

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By comparison theorem , $\"\"$ is also convergent.

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Therefore, Laplace transform $\"\"$ exists for every $\"\"$ .

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$\"\"$

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Laplace transform $\"\"$ exists for every $\"\"$ .