$\"\"$

\

(a)

\

The integral is $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

The integral is $\"\"$ .

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$ .

\

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

\

$\"\"$

\

(b)

\

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

\

Graph :

\

Graph of the function $\"\"$ .

\

$\"\"$

\

Observe the graph :

\

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

\

Therefore, $\"\"$ is converges.

\

$\"\"$

\

(c)

\

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

\

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

\

Consider $\"\"$ .

\

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

\

Consider $\"\"$ .

\

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

\

$\"\"$ .

\

$\"\"$

\

Consider $\"\"$ .

\

Integrate on each side.

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

$\"\"$

\

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

\

$\"\"$

\

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

\

(b)

\

Graph of the function $\"\"$ is

\

$\"\"$

\

Observe the graph :

\

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

\

Therefore, $\"\"$ is converges.

\

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