# AP Calculus AB : Asymptotic and Unbounded Behavior

## Example Questions

### Example Question #8 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Evaluate the following indefinite integral.

Explanation:

Use the inverse Power Rule to evaluate the integral.  We know that  for .  We see that this rule tells us to increase the power of  by 1 and multiply by   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 #9 : Comparing Relative Magnitudes Of Functions And Their Rates Of Change

Evaluate the following indefinite integral.

Explanation:

Use the inverse Power Rule to evaluate the integral.  Firstly, constants can be taken out of the integral, so we pull the 1/2 out front and then complete the integration according to the rule. We know that  for .  We see that this rule tells us to increase the power of  by 1 and multiply by .  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 #31 : Asymptotic And Unbounded Behavior

Evaluate the following indefinite integral.

Explanation:

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 .  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 #32 : Asymptotic And Unbounded Behavior

Evaluate the following indefinite integral.

Explanation:

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 #32 : Functions, Graphs, And Limits

Evaluate the following definite integral.

Explanation:

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 #31 : Asymptotic And Unbounded Behavior

Evaluate the following definite integral.

Explanation:

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 #32 : Asymptotic And Unbounded Behavior

Evaluate the following definite integral.

Explanation:

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 #251 : Ap Calculus Ab

Evaluate the following indefinite integral.

Explanation:

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 #252 : Ap Calculus Ab

Evaluate the following indefinite integral.

Explanation:

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 #32 : Asymptotic And Unbounded Behavior

Evalulate the following indefinite integral.