# Calculus 2 : Integrals

## Example Questions

### Example Question #881 : Integrals

Explanation:

To find this integral, look at each term separately.

For the first term, raise the exponent by 1 and also put that result on the denominator: .

For the next term, do the same: .

Same for the third term (since it's a constant, multiply the coefficient by x): .

Put those all together to get: . Since this is an indefinite integral, make sure to add C at the end: .

### Example Question #254 : Indefinite Integrals

Explanation:

First chop up the fraction into two separate terms and simplify: .

Now, integrate that expression. Remember to raise the exponent by 1 and also put that result on the denominator:

Since it's an indefinite integral, remember to add C at the end: .

### Example Question #881 : Integrals

Evaluate:

The integral does not converge

Explanation:

, so

### Example Question #1 : Improper Integrals

Evaluate:

Explanation:

First, we will find the indefinite integral using integration by parts.

We will let  and .

Then  and .

To find , we use another integration by parts:

, which means that , and

, which means that, again, .

Since

, or,

for all real , and

,

by the Squeeze Theorem,

.

### Example Question #1 : Improper Integrals

Evaluate:

The integral does not converge

Explanation:

First, we will find the indefinite integral, .

We will let  and .

Then,

and .

and

Now, this expression evaluated at is equal to

.

At it is undefined, because does not exist.

We can use L'Hospital's rule to find its limit as , as follows:

and , so by L'Hospital's rule,

Therefore,

### Example Question #882 : Integrals

Evaluate:

The integral does not converge.

Explanation:

First, we will find the indefinite integral

using integration by parts.

We will let  and .

Then  and .

Also,

.

To find , we can substitute  for . Then  or , so the antiderivative is

.

Now we can integrate by parts:

By L'Hospital's rule, since both the numerator and the denominator approach  as ,

.

So:

### Example Question #162 : Ap Calculus Bc

Evaluate:

Explanation:

Rewrite the integral as

.

Substitute . Then

and . The lower bound of integration stays , and the upper bound becomes , so

Since , the above is equal to

.

### Example Question #532 : Finding Integrals

Evaluate:

The integral does not converge.

The integral does not converge.

Explanation:

Substitute .

The lower bound of integration becomes ;

the upper bound becomes .

The integral therefore becomes

The integral, therefore, does not converge.

### Example Question #883 : Integrals

Evaluate the improper integral:

The integral does not converge.

Explanation:

First, we will perform an integration by parts on the indefinite integral:

Let  and

Then,

, ,

and

.

We do another integration by parts, setting

and .

Then,

and .

Also,

and, again, .

Therefore, the antiderivative of  is .

, which two applications of L'Hospital's rule can easily reveal.

Therefore,

.

### Example Question #884 : Integrals

Evaluate the improper integral:

The integral does not converge.

Explanation:

First, we will perform an integration by parts on the indefinite integral

.

Let  and

Then,

and .

Also,

.

Therefore,

The antiderivative of  is

and

.

, as can be proved by L'Hospital's rule.