Calculus 2 : Integrals

Study concepts, example questions & explanations for Calculus 2

varsity tutors app store varsity tutors android store

Example Questions

Example Question #21 : Definition Of Integral

Find the indefinite integral.

Possible Answers:

Correct answer:

Explanation:

Use Integration By Parts (IBP) to simplify the integral, using the IBP formula

Assign the variables  and  to appropriate expressions from the original integral,

Substitute these new expressions into the IBP formula and our integral becomes as follows,

                             

Use IBP again on the integral  and assign the variables  and  to appropriate expressions from the integral,

Substitute these new expressions into the IBP formula again and our full expression becomes as follows,

Integrate using the trig integration rules as follows,

So the final indefinite integral expression becomes

And the final answer is as follows,

Example Question #22 : Definition Of Integral

Find the indefinite integral.

Possible Answers:

Correct answer:

Explanation:

Make an appropriate u-substitution to simplify the integral.

Since ,

Substitute the new expressions containing  and  back into the original integral as follows,

Use partial fraction decomposition on the integral to simplify the expression as follows,

Set  then,

Set  then,

Set  then,

So,

And our integral expression becomes

Integrate each term. We will need to use the power rule and natural log integration as follows,

Where 

This results in our final indefinite integral,

Substitute back in our original u-substitution  and simplify to give us our final answer

Example Question #23 : Definition Of Integral

Find the definite integral.

Possible Answers:

Correct answer:

Explanation:

Simplify the integral using an appropriate u-substitution,

Since  ,

Substitute the new expressions containing  and  into the original integral as follows,

Use Integration By Parts (IBP) to simplify the integral further, using the IBP formula

Assign the variables  and  to appropriate expressions from the integral

Substitute these expressions back into the IBP formula and our integral becomes as follows,

Integrate using the trig integration rule

Our indefinite integral becomes

Substitute back in our original u-substitution  to get our final indefinite integral,

Apply the given boundaries  to  to get our final answer.

Example Question #21 : Integrals

Find the indefinite integral.

Possible Answers:

Correct answer:

Explanation:

Make an appropriate u-substitution to transform the integral into something we are familiar with.

Substitute the new expressions containing  and  into the original equation as follows,

Integrate each term using the power rule.

After integration we get the expression as follows,

Substitute back in our original u-substitution  to get the final answer.

Example Question #25 : Definition Of Integral

Find the indefinite integral.

Possible Answers:

Correct answer:

Explanation:

First use Integration By Parts (IBP) to make the integral doable, using the IBP formula,

Assign appropriate expressions from the original integral to the variables  and  to use in the IBP formula.

Substitute these expressions into the IBP formula and our integral becomes as follows,

Use long division to simplify the expression ,

           

        

So,

Substitute this back into its appropriate place in our expression and it becomes as follows,

Integrate each term using the power rule and natural log integration as follows,

Our indefinite integral becomes as follows,

Simplifying, our answer is

Example Question #26 : Definition Of Integral

Find the indefinite integral.

Possible Answers:

Correct answer:

Explanation:

First divide the numerator by the denominator, using long division.

                         

                        

Now we can write our integral as follows.

From here we will use partial fraction decomposition.

Set  then,

.

Set  then, 

Therefore we get,

.

Now we can rewrite our integral as follows,

.

From here integrate each term. To integrate we will need to use the power rule, integration of a constant, and nautral log integration we are as follows.

Resulting in our final answer,

.

Example Question #27 : Definition Of Integral

Find the indefinite integral.

Possible Answers:

Correct answer:

Explanation:

Simplify the integrand using long division.

               

          

The integral becomes

Use partial fraction decomposition to simplify the expression ,

Set  then,

Set  then,

Set  then,

Therefore we get

Now we can rewrite our integral as follows,

Integrate each term using the power rule, integration of a constant, and natural log integration as follows,

Resulting in our final answer,

Example Question #28 : Definition Of Integral

Find the indefinite integral.

Possible Answers:

Correct answer:

Explanation:

Use Integration By Parts (IBP) to make the integral doable, using the IBP formula.

Assign appropriate expressions from the original integral to the variables  and  to use in the IBP formula.

Substitute these expressions into the IBP formula and our integral becomes as follows,

Use partial fraction decomposition to further simplify 

Set  then,

Set  then,

So,

Therefore we get

Finish the integration using the natural log integration rule

Resulting in the final answer,

Example Question #22 : Integrals

Find the indefinite integral.

Possible Answers:

Correct answer:

Explanation:

Simplify the integral choosing an appropriate u-substitution

Substitute the new expressions containing  back into the original integral as follows,

Since ,

Integrate using the power rule of integration,

Resulting in the answer after integrating,

Substitute back in our original  for our final answer,

Example Question #30 : Definition Of Integral

Find the indefinite integral.

Possible Answers:

Correct answer:

Explanation:

Make an appropriate u-substitution to turn the integral into something we are familiar with,

.

And since  ,

Substitute the new expressions containing  back into the original integral as follows,

.

From here we will use Integration By Parts (IBP) to simplify the integral further, using the IBP formula

.

Assign the variables  and  to appropriate expressions from the original integral,

.

Substitute these new expressions back into the IBP formula and our integral becomes as follows,

.

Integrate using the following integration rule:

.

This results in the expression

.

Substitute back in our original u-substitution ,

.

Simplify to get the final answer,

.

Learning Tools by Varsity Tutors