$\"\"$

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The integral is $\"\"$ .

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Definition of an improper integral :

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If $\"\"$ exists for every number $\"\"$ , then $\"\"$ provoded this limit exists (as a finite number).

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Consider $\"\"$ .

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Let $\"\"$ , then $\"\"$ .

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Differentiate on each side with respect to $\"\"$ .

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$\"\"$

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$\"\"$

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Boundaries : from $\"\"$ to $\"\"$ .

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$\"\"$

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Substitute corrresponding values in the integral function.

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$\"\"$

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$\"\"$

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$\"\"$

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The trigonometryic identity : $\"\"$

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$\"\"$

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$\"\"$

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Therefore, $\"\"$ .

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Since $\"\"$ (finite value), the integral is convergent. $\"\"$

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The integral is convergent and the value is $\"\"$ .