AP Calculus AB : Antiderivatives following directly from derivatives of basic functions

Study concepts, example questions & explanations for AP Calculus AB

varsity tutors app store varsity tutors android store

Example Questions

Example Question #36 : Techniques Of Antidifferentiation

Calculate the integral in the following expression:

Possible Answers:

Correct answer:

Explanation:

Solving this integral depends on knowledge of exponent rules; mainly, that . Using this, we can simplify the given expression.

 

From here, we just follow the power rule, raising the exponent and dividing by it.

Giving us our final answer.

Example Question #37 : Techniques Of Antidifferentiation

Evaluate the integral

Possible Answers:

Correct answer:

Explanation:

To evaluate the integral, we use the following definition

Example Question #38 : Techniques Of Antidifferentiation

Evaluate the following integral

Possible Answers:

Correct answer:

Explanation:

To solve the problem, we apply the fact that anti-derivative of  and that 

Taking the anti-derivative of each part independently, we get

Finally, our answer is

 

Example Question #39 : Techniques Of Antidifferentiation

Determine the value of \int_{0}^{2}\sqrt{9x^2+12x+4} dx.

Possible Answers:

Correct answer:

Explanation:

We can factor the equation inside the square root:

From here, increase each term's exponent by one and divide the term by the new exponent.

Now, substitute in the upper bound into the function and subtract the lower bound function value from it.

Therefore,

\int_{0}^{2}\sqrt{(3x+2)^2}dx=\int_{0}^{2}(3x+2)dx=10

Example Question #102 : Integrals

Evaluate the following integral

Possible Answers:

Correct answer:

Explanation:

To evaluate the integral, we use the definition 

Example Question #31 : Antiderivatives Following Directly From Derivatives Of Basic Functions

Evaluate the following integral

Possible Answers:

Correct answer:

Explanation:

To evaluate the integral, we use the fact that the antiderivative of  is  (because ), and the antiderivative of  is  (because ). Using this information, we determine that the integral is

Example Question #32 : Antiderivatives Following Directly From Derivatives Of Basic Functions

Calculate the following integral.

Possible Answers:

Correct answer:

Explanation:

Calculate the following integral.

To do this problem, we need to recall that integrals are also called antiderivatives. This means that we can calculate integrals by reversing our integration rules.

Thus, we can have the following rules.

Using these rules, we can find our answer:

 

Will become:

And so our answer is:

Example Question #33 : Antiderivatives Following Directly From Derivatives Of Basic Functions

Integrate:

Possible Answers:

Correct answer:

Explanation:

The integral of the function is equal to

and was found using the following rule:

Finally, we evaluate by plugging in the upper bound into the resulting function and subtracting the resulting function with the lower bound plugged in:

Example Question #34 : Antiderivatives Following Directly From Derivatives Of Basic Functions

Solve:

Possible Answers:

Correct answer:

Explanation:

The integral is equal to

The rules used to integrate are

Now, we solve by plugging in the upper bound of integration and then subtracting the result of plugging in the lower bound of integration:

 

 

Example Question #35 : Antiderivatives Following Directly From Derivatives Of Basic Functions

Integrate:

Possible Answers:

Correct answer:

Explanation:

The integral is equal to

and was found using the following rule:

Learning Tools by Varsity Tutors