AP Calculus AB : Use of the Fundamental Theorem to evaluate definite integrals
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Example Question #31 : Fundamental Theorem Of Calculus
Evaluate the integral 
The integral has a group raised to a power. This usually follows the basic integral form 
Before we try to apply this, we should move the variables in the denominator up to the top. For our integral,
Now we try u-substitution, with the group 
Our 
Now we can replace all the "x-stuff" with "u-stuff".
Notice that the integral bounds, 
To find the equivalent u bounds, we take the u substitution we used earlier, 
For 
Now for 
Having found the u-bounds, we will now begin to integrate. First we will pull the 
Now we use the power rule for integration, 
The 
Now we apply the 2nd Fundamental Theorem of Calculus, 
For us, 
Applying this theorem, we get
This is the correct answer.
Example Question #22 : Use Of The Fundamental Theorem To Evaluate Definite Integrals
Evaluate the integral
To evaluate this integral, we must first rewrite the 
Since the 
Now we find 
This gives us the cosine piece. Writing everything in terms or u, we get
To integrate this we use the following basic integral form:
Applying this to our integral, we get
Now we can "un-substitute" to get back to x-terms. Replacing 
Now we use the 2nd Fundamental Theorem of Calculus. We plug in the upper bound of 
Doing this we get
This is the correct answer.
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