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February 4th, 2016, 12:59 AM  #1 
Member Joined: Jan 2016 From: / Posts: 43 Thanks: 1  Fundamental theorem of calculus question.
here : I just stumbled upon this khan academy video, explaining fundamental theorem of calculus. I am sure than mr. Khan got the right result, but I'm just curious, Couldn't we just swap the bounds of the given expression, get the negative form and evaluate it with the help of the fundamental theorem of calculus? I tried to evaluate it and got a different result, so my question is why this doesn't work? thanks! 
February 4th, 2016, 02:33 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,689 Thanks: 2669 Math Focus: Mainly analysis and algebra 
Can you write out the expression?

February 4th, 2016, 04:23 AM  #3 
Member Joined: Jan 2016 From: / Posts: 43 Thanks: 1 
I can try! here I hope it's attached right! 
February 4th, 2016, 07:54 AM  #4 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,689 Thanks: 2669 Math Focus: Mainly analysis and algebra 
I had to watch the video to try to understand what you mean, and I'm still not sure. Are you suggesting that we write: $$F(x) = \int \limits_x^{x^2} {\cos t \over t} \, \mathrm d t = \int \limits_{x^2}^x {\cos t \over t} \, \mathrm d t$$ and then try to apply the Fundamental Theorem of Calculus to the final expression? The point about the Fundamental Theorem of Calculus is that says that if $$F(x) = \int \limits_a^x f(t) \, \mathrm d t$$ then $$F'(x) = f(x)$$ for any constant $a$. In this example, we don't have a constant limit  both limits are in the variable $x$, so we have to introduce the constant in order to use the theorem. 
February 4th, 2016, 08:03 AM  #5 
Member Joined: Jan 2016 From: / Posts: 43 Thanks: 1 
Exactly what I meant! Sure.. the thing breaks at the constant. And you're right x is a variable! It all makes sense. Thanks once again! 
February 4th, 2016, 03:25 PM  #6 
Global Moderator Joined: Oct 2008 From: London, Ontario, Canada  The Forest City Posts: 7,968 Thanks: 1152 Math Focus: Elementary mathematics and beyond  

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