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

Example Questions

Example Question #26 : Techniques Of Antidifferentiation

Integrate:

Explanation:

The integral of the function is equal to

The rules used for integration were

For the definite component of the integration, we plug in the upper limit of integration, and subtract the result of plugging in the lower limit of integration:

Example Question #27 : Techniques Of Antidifferentiation

Evaluate the integral

Explanation:

To find the derivative of the expression, we use the following rule

Applying to the integrand from the problem statement, we get

Example Question #28 : Techniques Of Antidifferentiation

Find the antiderivative of the following.

Explanation:

Follow the following formula to find the antiderivatives of exponential functions:

Thus, the antiderivative of  is .

Example Question #29 : Techniques Of Antidifferentiation

Find the antiderivative of the following.

Explanation:

is the derivative of . Thus, the antiderivative of  is .

Example Question #30 : Techniques Of Antidifferentiation

Find the antiderivative of the following.

Explanation:

is the derivative of . Thus, the antiderivative of  is .

Example Question #31 : Techniques Of Antidifferentiation

Define

Evaluate .

Explanation:

has different definitions on  and , so the integral must be rewritten as the sum of two separate integrals:

Calculate the integrals separately, then add:

Example Question #21 : Antiderivatives Following Directly From Derivatives Of Basic Functions

Evaluate the integral

Explanation:

To evaluate the integral, we use the rules for integration which tell us

Applying to the integral from the problem statement, we get

Example Question #33 : Techniques Of Antidifferentiation

Integrate:

Explanation:

To evaluate the integral, we can split it into two integrals:

After integrating, we get

where a single constant of integration comes from the sum of the two integration constants from the two individual integrals, added together.

The rules used to integrate are

Example Question #34 : Techniques Of Antidifferentiation

Solve:

Explanation:

The integral is equal to

and was found using the following rule:

where

Example Question #35 : Techniques Of Antidifferentiation

Solve:

Explanation:

To integrate, we can split the integral into the sum of two separate integrals:

Integrating, we get

which was found using the following rules:

Note that the constants of integration were combined to make a single integration constant in the final answer.

(The first integral can be rewritten as   for clarity.)