All AP Calculus AB Resources
Example Questions
Example Question #11 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change
Evaluate the following indefinite integral.
Use the inverse Power Rule to evaluate the integral. We know that for . But, in this case, IS equal to so a special condition of the rule applies. We must instead use . Pull the constant "3" out front and evaluate accordingly. Next always add your constant of integration that would be lost in the differentiation. Take the derivative of your answer to check your work.
Example Question #12 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change
Evaluate the following definite integral.
Unlike an indefinite integral, the definite integral must be evaluated at its limits, in this case, from 0 to 2. First, we use our inverse power rule to find the antiderivative. So, we have that . Once you find the antiderivative, we must remember that where is the indefinite integral. So, we plug in our limits and subtract the two. So, we have .
Example Question #41 : Functions, Graphs, And Limits
Evaluate the following definite integral.
Unlike an indefinite integral, the definite integral must be evaluated at its limits, in this case, from 1 to 3. First, we use our inverse power rule to find the antiderivative. So, we have that . Once you find the antiderivative, we must remember that where is the indefinite integral. So, we plug in our limits and subtract the two. So, we have .
Example Question #42 : Functions, Graphs, And Limits
Evaluate the following definite integral.
Unlike an indefinite integral, the definite integral must be evaluated at its limits, in this case, from 1 to 4. First, we use our inverse power rule to find the antiderivative. So since is to the power of , we have that . Once you find the antiderivative, we must remember that where is the indefinite integral. So, we plug in our limits and subtract the two. So, we have because we know that .
Example Question #43 : Functions, Graphs, And Limits
Evaluate the following indefinite integral.
First, we know that we can pull the constant "4" out of the integral, and we then evaluate the integral according to this equation:
. From this, we acquire the answer above. As a note, we cannot forget the constant of integration which would be lost during the differentiation.
Example Question #13 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change
Evaluate the following indefinite integral.
First, we know that the integral of a sum is the same as the sum of the integrals, so if needed, we can split the three integrals up and evaluate them seperately. We then evaluate each integral according to this equation:
. From this, we acquire the answer above. As a note, we cannot forget the constant of integration which would be lost during the differentiation.
Example Question #41 : Functions, Graphs, And Limits
Evalulate the following indefinite integral.
Normally, we would evalute the indefinite integral according to the following equation:
. However, in this case, . Now we use our other rule that states the integral of is equal to plus a constant. From this, we acquire the answer above. As a note, we cannot forget the constant of integration which would be lost during the differentiation.
Example Question #41 : Functions, Graphs, And Limits
Evaluate the following indefinite integral.
We evaluate the integral according to this equation:
. From this, we acquire the answer above. Keep in mind that is the same as . As a note, we cannot forget the constant of integration which would be lost during the differentiation.
Example Question #15 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change
Evaluate the following indefinite integral.
First, we remember that the integral of a sum is the same as the sum of the integrals, so we can split the sum into seperate integrals and solve them individually. We then evaluate each integral according to this equation:
. From this, we acquire the answer above. As a note, we cannot forget the constant of integration which would be lost during the differentiation.
Example Question #16 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change
Evaluate the following indefinite integral.
First, we know that we can pull the constant out of the integral, and we then evaluate the integral according to this equation:
. From this, we acquire the answer above. As a note, we cannot forget the constant of integration which would be lost during the differentiation.