Calculus 2 : Integrals

Study concepts, example questions & explanations for Calculus 2

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Example Questions

Example Question #11 : Solving Integrals By Substitution

Evaluate the integral:

Possible Answers:

Correct answer:

Explanation:

To evaluate the integral, we must recognize that what we were given looks very similar to the following integral:

To make our integral look like the one above, we must perform the following substiution:

Now, rewrite our integral:

It looks like the one above, so we can integrate now:

Finally, replace u with our original term:

Example Question #2701 : Calculus Ii

Evaluate the following integral:

Possible Answers:

Correct answer:

Explanation:

To integrate, we must break the integral into three integrals:

The first integral is equal to 

and was found using the following rule:

The second integral is equal to

and was found using the following rule:

The final integral is found by performing the following substitution:

Now, rewrite and integrate:

The integral was found using the following rule:

Finally, rewrite the integral in terms of  by replacing  with the original term, and add all three integrals together to get a final answer of 

Example Question #12 : Solving Integrals By Substitution

Evaluate the following integral:

Possible Answers:

Correct answer:

Explanation:

To integrate, we must make the following substitution:

The derivative was found using the following rule:

Now, rewrite the integral:

Notice that we changed

Next, distribute and integrate:

The integral was found using the following rule:

Finally, replace  with our original  term:

Example Question #20 : Solving Integrals By Substitution

Evaluate the following integral:

Possible Answers:

Correct answer:

Explanation:

To solve the integral, we must make the following substitution:

Now, rewrite the integral and integrate:

The integral was found using the following rule:

Finally, replace  with our original  term:

Example Question #21 : Solving Integrals By Substitution

Evaluate the following integral:

Possible Answers:

Correct answer:

Explanation:

To evaluate the integral, we first must rewrite it as the following:

Now, perform the following subsitution:

Next, rewrite the integral and integrate:

The integral was performed using the following rule:

Finally, replace  with the  term:

 

Example Question #21 : Solving Integrals By Substitution

Evaluate the following integral:

Possible Answers:

Correct answer:

Explanation:

To evaluate the integral, we must first make the following substitution:

Now, rewrite the integral and integrate:

The integral was performed using the following rule:

Finally, replace u with the x term:

 

 

Example Question #22 : Solving Integrals By Substitution

Evaluate the following integral:

Possible Answers:

Correct answer:

Explanation:

To integrate, we must first make the following substitution:

Now, rewrite the integral in terms of u, and integrate:

The integration was performed using the following rule:

Finally, replace u with our original term:

Example Question #23 : Solving Integrals By Substitution

Evaluate the following integral:

Possible Answers:

Correct answer:

Explanation:

To evaluate the integral, we first must make the following substitution:

Now, rewrite the integral, and integrate:

We used the following rule for integration:

Finally, replace  with our original term:

Example Question #23 : Solving Integrals By Substitution

Evaluate the following integral:

Possible Answers:

Correct answer:

Explanation:

To evaluate the integral, first we must make the following substitution:

The derivative was found using the following rule:

Now, rewrite the integral and integrate:

The integral was performed using the following rule:

Finally, replace  with the  containing term:

Note that we removed the absolute value sign because the output of a square root is always positive.

Example Question #26 : Solving Integrals By Substitution

Evaluate the following integral:

Possible Answers:

 

Correct answer:

Explanation:

To evaluate the integral, we must split it into two integrals:

The first integral is equal to

and was found using the following rule:

The second integral is solved by performing the following substitution:

Now, rewrite the integral and integrate:

The integration was performed using the following rule:

Finally, replace  with our original term and add the two results of the integrations together:

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