Calculus 2 : Integrals

Study concepts, example questions & explanations for Calculus 2

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Example Questions

Example Question #1 : Solving Integrals By Substitution

Solve the following indefinite integral:

Possible Answers:

Correct answer:

Explanation:

To solve the following integral, we must make a substitution to create the following general form:

We make the following subsitution:

The derivative was found using the following rule:

The integral now looks like this:

Notice that it is in the same form as the integral we want.

Now use the form from above to integrate:

To finish the problem, replace u with 2x:

.

Example Question #2691 : Calculus Ii

Simplify the following indefinite integral.

Possible Answers:

Correct answer:

Explanation:

We can simplify

 

by first doing a substitution, with , which gives us , which means that . So the integral becomes

The integral can be solved using two integration by parts, which will give us 

So now we just plg in  into  and  into  to get

Example Question #1 : Solving Integrals By Substitution

Calculate the integral:  

Possible Answers:

 

Correct answer:

 

Explanation:

Pull out the constant out in front of the integral.

Use U-substitution to solve.

Example Question #1 : Solving Integrals By Substitution

Solve the following indefinite integral:

Possible Answers:

Correct answer:

Explanation:

The integral can be solved with a clever substitution:

The derivative was found using the following rules:

,

Then, when you rewrite the integral in terms of u, you find that you get:

The integration was performed using the following rule:

Finally, replace the u with our original term.

Example Question #11 : Solving Integrals By Substitution

Solve the following indefinite integral:

Possible Answers:

Correct answer:

Explanation:

The integral is found by recognizing the following integral:

To solve the integral, make the following substitution:

The derivative was found using the following rule:

The integral then becomes

Finish by replacing u with the original term containing x.

Example Question #12 : Solving Integrals By Substitution

Evaluate the following integral:

 

Possible Answers:

Correct answer:

Explanation:

Substitution can be used to make this problem easier. Let u = sin(x). Then du = cos(x)dx. This allows us to rewrite and evaluate the original integral in terms of u as:

The final answer should be written in terms of the original variables, so substitute sin(x) for u. 

 

Note that we could have also chosen cos(x) as u, but the above substitution avoids introducing negatives. 

Example Question #13 : Solving Integrals By Substitution

Evaluate the integral .

Possible Answers:

Correct answer:

Explanation:

First, notice that . Use the substitution  to rewrite the integral as

.

Next, recall that , so 

 

Lastly, substitute  in place of  to write the answer in terms of the original variables. 

 

Example Question #14 : Solving Integrals By Substitution

Solve the following indefinite integral:

Possible Answers:

Correct answer:

Explanation:

To solve the indefinite integral, make a simple substitution:

The integral then becomes:

After integrating, we get

The following rule was used for the integration:

Finally, replace the u with the original term containing x. 

Example Question #15 : Solving Integrals By Substitution

Please solve the following integral:

Possible Answers:

    

Correct answer:

    

Explanation:

We know that the derivative of   is  .

Doing a substitution and setting

 and  allows us to rewrite the integral as 

 which can be rewritten as .

Integrating this gets you  plus a constant (which is stated in the original question that you can assume that we already have one). Substituting  back in gives us the final answer, which is 

.

Example Question #16 : Solving Integrals By Substitution

Please solve the following integral. 

  

Possible Answers:

Correct answer:

Explanation:

We know that the derivative of  is .

So substituting

 allows us to have 

.

This allows us to rewrite the integral as 

 which, when integrated, gives us 

.

Substiting x back in gives us the answer, 

.

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