Calculus 2 : Integrals

Study concepts, example questions & explanations for Calculus 2

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Example Questions

Example Question #881 : Integrals

Possible Answers:

Correct answer:

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

Possible Answers:

Correct answer:

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:

Possible Answers:

The integral does not converge

Correct answer:

Explanation:

, so

 

Example Question #1 : Improper Integrals

Evaluate:

Possible Answers:

Correct answer:

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:

Possible Answers:

The integral does not converge

Correct answer:

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:

 

Possible Answers:

The integral does not converge.

Correct answer:

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:

Possible Answers:

Correct answer:

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:

Possible Answers:

The integral does not converge.

Correct answer:

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:

Possible Answers:

The integral does not converge.

Correct answer:

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:

Possible Answers:

The integral does not converge.

Correct answer:

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.

 

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