Calculus 2 : Integrals

Study concepts, example questions & explanations for Calculus 2

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Example Questions

Example Question #971 : Integrals

Evaluate the following integral:

Possible Answers:

Correct answer:

Explanation:

To integrate, we must first make the following substitution:

Next, rewrite the integral and integrate:

The integration was performed using the following rule:

Finally, replace u with our original x term:

Example Question #37 : Solving Integrals By Substitution

What is the integral of the following equation?

Possible Answers:

Correct answer:

Explanation:

We can solve this integral with u substitution

 let , so , or, 

Making this substitution, and moving our constants gives us:

, solving the integral, we get , plugging our value for u back into the equation 

Example Question #38 : Solving Integrals By Substitution

Possible Answers:

Correct answer:

Explanation:

To make this integral simpler, we will need to make a substitution.  You want to pick a substitution where the derivative also exists in the integral.  Here, we want to choose:

.  Now, we want to rewrite the integral interms of the new variable.

.  

The last step is just to substitute the original substitution back in. 

.

 

Example Question #32 : Solving Integrals By Substitution

7q

Possible Answers:

Correct answer:

Explanation:

7a

Example Question #41 : Solving Integrals By Substitution

Solve the indefinite integral using trigonemtric substitution

Possible Answers:

Correct answer:

Explanation:

We substitute 

to solve the integral. Solving for dx,

Substituting these values into the integral yields

Solving for  from

gives us

And so the indefinite integral is

Example Question #42 : Solving Integrals By Substitution

Evaluate the following indefinite integral:

Possible Answers:

Correct answer:

Explanation:

The integrand is composed of a function as well as its derivative multiplied by a constant. Hence, we can find the antiderivative via u-substitution as follows:

Let . Then , and so . Thus,

 

                                  

                                  

Example Question #43 : Solving Integrals By Substitution

Evaluate the following indefinite integral:

Possible Answers:

Correct answer:

Explanation:

The integrand can be evaluated by means of the u-substitution method, as follows:

Let . Then , and so

Example Question #44 : Solving Integrals By Substitution

Evaluate the following indefinite integral:

Possible Answers:

Correct answer:

Explanation:

Here, an understanding of trigonometric identities, as well as the appropriate selection of a dummy variable for u-substitution, is required. To figure out which function to represent "u" (cosine or sine), simply re-write the integrand as

                                

Remembering that ,

Now, we can substitute  to yield

 

because if , then , which implies .

At this point, all that is left to do is expand the polynomial and evaluate the integrand:

                               

                                

Example Question #45 : Solving Integrals By Substitution

Find the value of

.

Possible Answers:

Correct answer:

Explanation:

To perform this integration, we use a substitution.

Since the derivative of is , we choose our substitution to be .

Differentiating gives us,

 .

Now we can substitute this into our integral. We will have,

 .

Along with this substitution, we must also change our limits of and . To do so, we take these values and plug them in for  in the formula .

Doing so, we obtain and .

Now our integral will be transformed as follows,

 .

This integral is now easy to integrate, for the function integrates to .

Thus we have,

 .

Therefore, the answer to the integral is,

.

Example Question #44 : 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 integral was performed using the following rule:

Note that the rule contains a fraction in front of the inverse trig function. Do not confuse this fraction with the fraction coming from the u substitution!

Finally, replace u with our original term and multiply the constants:

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