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

## Example Questions

### Example Question #36 : Techniques Of Antidifferentiation

Calculate the integral in the following expression:

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.

### Example Question #37 : Techniques Of Antidifferentiation

Evaluate the integral

Explanation:

To evaluate the integral, we use the following definition

### Example Question #38 : Techniques Of Antidifferentiation

Evaluate the following integral

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

### Example Question #39 : Techniques Of Antidifferentiation

Determine the value of .

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,

### Example Question #102 : Integrals

Evaluate the following integral

Explanation:

To evaluate the integral, we use the definition

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

Evaluate the following integral

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.

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:

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

Integrate:

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:

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:

Integrate: