Calculus 1 : How to find integral expressions

Study concepts, example questions & explanations for Calculus 1

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Example Questions

Example Question #71 : Equations

Find the indefinite integral.

Possible Answers:

None of these

Correct answer:

Explanation:

For this integral you need to know an integration rule for each part of the problem.

 

Using this rule we can get the integral of the first part

.

For the second part we must know that

so

.

and finally the last part follows the rule exactly

.

We must also include a C for the constant of integration as the integral is indefinite.

Thus the final answer is

.

Example Question #71 : Equations

Give the integral expression for the velocity of an object falling from initial velocity  towards the earth with acceleration , between  and  seconds after it was released. 

Possible Answers:

Correct answer:

Explanation:

We know that the definite integral of acceleration between 0 and 4 seconds is the velocity. We know that the acceleration due to gravity is . We also know that the inititial velocity is . This means that when . Therefore, the velocity function can be written as 

Example Question #71 : Equations

Evaluate the following integral:

Possible Answers:

Correct answer:

Explanation:

To integrate, we must make the following substitution:

We used the following rule to derivate:

Now, rewrite the integral and integrate:

We used the following rule for the integration:

Now, replace u with the original x-containing term to finish:

 

Example Question #74 : Equations

Evaluate the following integral:

Possible Answers:

Correct answer:

Explanation:

To integrate, we must make the following subsitution:

We used the following rule to derivate:

Now, after rearranging, we get the following integral:

and after integrating we get

We used the following rule to integrate:

To finish, we plug our x term back in place of u:

Example Question #75 : Equations

Evaluate the following integral:

Possible Answers:

Correct answer:

Explanation:

After integrating, we get

The integration was performed using the following rules:

Example Question #76 : Equations

Evaluate the following integral:

Possible Answers:

Correct answer:

Explanation:

In order to evaluate this integral, we must split the integrand into two seperate integrals:

For the first integral, we need to make the following substitution:

We found the derivative using the following rule:

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

We integrated using the following rule:

For the second integral, we just integrate:

and we use the rule

Now, just add together the results of the two integrals (the integral of the sum is equal to the sum of the integrals):

Example Question #71 : Writing Equations

Evaluate the following integral:

Possible Answers:

Correct answer:

Explanation:

To integrate, we must make the following substitution:

We found the derivative using the following rule:

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

We used the following integration rule:

Now, replace u with the original term containing x:

Example Question #78 : Equations

Evaluate the following integral:

Possible Answers:

Correct answer:

Explanation:

When we integrate, we get

which was found using the rule

Example Question #1122 : Functions

Evaluate the following indefinite integral:

 

Possible Answers:

Correct answer:

Explanation:

To evaluate the integral the integral, use the inverse power rule:

Applying that rule to this problem gives us the following for the first term:

The following for the second term:

And the following for the third term:

We can combine these terms and add our "C" to get the final answer:

Example Question #80 : Equations

Evaluate the indefinite integral:

Possible Answers:

Correct answer:

Explanation:

To evaluate this integral, we have to remember the trig laws. When taking the integral of cosine, our answer is always sine. In this case we do the following:

Move the 2 outside of the integral:

Evaluate the integral of cosine and add our "C"

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