Fundamental theorem of calculus

  • #1
e^(i Pi)+1=0
247
1
[itex]\frac{d}{dx} \int_a^b f(x) dx=f(b)[/itex]

This is something I can churn through mechanically but I never "got." Any links / explanations that can help build my intuition about this would be helpful.
 

Answers and Replies

  • #2
36,709
8,708
[itex]\frac{d}{dx} \int_a^b f(x)=f(b)[/itex]

This is something I can churn through mechanically but I never "got." Any links / explanations that can help build my intuition about this would be helpful.

What you have is incorrect, assuming that both a and b are constants.
[tex]\frac{d}{dx} \int_a^b f(x) dx =0[/tex]

The way this is usually presented is like so:

[tex]\frac{d}{dx} \int_a^x f(t) dt =f(x)[/tex]
 
  • #3
36,709
8,708
For an explanation, let's assume that F(x) is an antiderivative of f(x). IOW, F'(x) = f(x).
Then
$$ \int_a^x f(t) dt = F(x) - F(a)$$
So $$ d/dx \int_a^x f(t) dt = d/dx( F(x) - F(a)) = F'(x) - 0 = f(x)$$
 

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