$\"\"$

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The integral is $\"\"$ .

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$\"\"$ .

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If $\"\"$ is continuous on the interval $\"\"$ then $\"\"$ is convergent.

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Solve the integral by using integration by parts of integration.

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Apply integration by parts formula $\"\"$ .

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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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$\"\"$

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$\"\"$

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$\"\"$

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$\"\"$

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Apply the limits.

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$\"\"$

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$\"\"$

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Apply L hospital rule:

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$\"\"$

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$\"\"$

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$\"\"$

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Therefore, the integral $\"\"$ converges to $\"\"$ . $\"\"$

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The improper integral $\"\"$ converges to $\"\"$ .