Problem 1 asks for the partial fractions decomposition of To find this, first we factor the denominator:

This means that there must be constants such that

To find these constants, we add the fractions on the right hand side, and obtain

This means that

Setting gives us

Setting gives us

Setting gives us

Setting and using the values of just found, gives us

Putting this together, the partial fractions decomposition is

Problem 2 asks to determine whether converges. Note that the expression we are integrating is defined in but not at so this is an improper integral of Type II and to evaluate it we use the definition:

To evaluate this last expression, we use the result from Problem 1:

This expression diverges because but

Problem 3 asks to determine whether converges. Note that the expression we are integrating is defined in but, of course, the interval of integration is infinite, so this is an improper integral of Type I and to evaluate it we use the definition:

To evaluate this last expression, we proceed as in Problem 2:

To evaluate the expression within the first set of parentheses, we use that

The limit of this expression as is because using l’Hôpital’s rule. This means that

(In particular, the integral converges.)

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