$\"\"$

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(a)

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The integral is $\"\"$ .

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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 integral is $\"\"$ .

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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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$\"\"$ .

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Therefore, $\"\"$ is converges if $\"\"$ and the limit is diverges if $\"\"$ .

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$\"\"$

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(b)

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The function is $\"\"$ and the interval is $\"\"$ .

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Graph :

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Graph of the function $\"\"$ .

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$\"\"$

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Observe the graph:

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The function $\"\"$ is converges in the interval $\"\"$ .

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Therefore, $\"\"$ is converges.

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$\"\"$

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(c)

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Solve the $\"\"$ by using integration by parts.

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

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Consider $\"\"$ .

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Here $\"\"$ 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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Consider $\"\"$ .

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Integrate 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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Therefore, the integral is $\"\"$ is converges.

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$\"\"$

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(a) $\"\"$ is converges if $\"\"$ and the limit is diverges if $\"\"$ .

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(b)

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Graph of the function $\"\"$ is

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$\"\"$

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Observe the graph:

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The function $\"\"$ isconverges in the interval $\"\"$ .

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Therefore, $\"\"$ is converges.

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(c) The integral is $\"\"$ is converges.