help? [improper integrals]

Question

Answer 1

1.

Step 1 :

Comparison theorem :

Suppose that are continuous functions with ,

1. If is convergent, then is convergent.

2. If is divergent, then is also divergent.

Step 2 :

The integral is .

Now we can apply comparison theorem for above integral.

Consider the fact and it implies that  .

Now .

Here and .

The limit exists and is finite, so the integral is convergent.

Since is a finite value, it is convergent.

By comparison theorem, is also convergent.

Thus, is convergent.

Solution :

is convergent.

Answer 2

2.

Step 1 :

The integral is .

Now we can apply comparison theorem for above integral.

Consider the fact and it implies that .

Now . 

Here and .

The limit exists and is finite, so the integral is convergent.

Since is a finite value, it is convergent.

By comparison theorem, is also convergent.

Thus,  is convergent.

Solution :

 is convergent.

Answer 3

3.

Step 1 :

The integral is .

The limit exists and is finite, so the integral is convergent.

Thus, is convergent.

Solution :

is convergent.