All AP Calculus AB Resources
Example Questions
Example Question #51 :Asymptotic And Unbounded Behavior
Evaluate:
The antiderivative of. The derivative of. However, since there is no 2 in the original integral, we must divideby 2. Therefore, the answer is
Example Question #31 :Comparing Relative Magnitudes Of Functions And Their Rates Of Change
Evaluate the integral:
When taking the antiderivative add one to the exponent and then divide by the exponent.
Example Question #32 :Comparing Relative Magnitudes Of Functions And Their Rates Of Change
Evaluate the integral:
Cannot be evaluated
The derivative of. Therefore, the antiderivative ofis equal to itself.
Example Question #61 :Functions, Graphs, And Limits
Evaluate:
Can't be determined from the information given.
and
Recall thatis an odd function andis an even function.
Thus, sinceis an odd function, the integral of this function fromtowill be zero.
Example Question #61 :Functions, Graphs, And Limits
Evaluate this indefinite integral:
To approach this problem, first rewrite the integral expression as shown below:
.
Then, recognize that, and substitute this into the integral expression:
Use substitution, lettingand. The integral can then be rewritten as
Evaluating this integral gives
.
Finally, substitutingback into this expression gives the final answer:
(As this is an indefinite integral,must be included).
Example Question #1 :Finding Integrals
Evaluate:
Example Question #2 :Finding Integrals
Find
This is most easily solved by recognizing that.
Example Question #3 :Finding Integrals
Remember the fundamental theorem of calculus!
Since our, we can't use the power rule. Instead we end up with:
Remember to include thefor any anti-derivative or integral taken!
Now we can plug that equation into our FToC equation:
Notice that the c's cancel out. Plug in the given values for a and b and solve:
Example Question #31 :Integrals
The integral ofis. The constant 3 is simply multiplied by the integral.
Example Question #32 :Integrals
To integrate, we need to get the two equations in terms of each other. We are going to use "u-substitution" to create a new variable,, which will equal.
Now, if, then
Multiply both sides byto get the more familiar:
Note that our, and our original equation was asking for a positive.
That means if we wantin terms of, it looks like this:
Bring the negative sign to the outside:
.
We can use the power rule to find the integral of:
Since we said that, we can plug that back into the equation to get our answer:
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