##  [Fundamental Theorem of Calculus](/fundamental-theorem-calculus) 

  ##  [Fundamental Theorem of Calculus](https://mathlogic.quantumdictionary.io/fundamental-theorem-calculus-0) 

  

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- Natural & Formal Sciences -

**Mathematics & Logic Dictionary**

 







 

 

 

 



 

 

 

 

Definition

The pair of related results that link differentiation and definite integration: (1) if f is integrable on [a,b] and F(x)=∫_a^x f(t) dt then F is an antiderivative of f where f is continuous; (2) if F is any antiderivative of an integrable function f on [a,b], then ∫_a^b f(x) dx = F(b)−F(a).

 

 

 

 

 





 

 



 ##  [Fundamental Theorem of Calculus](https://puremath.quantumdictionary.io/fundamental-theorem-calculus-1) 

  

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- Mathematics & Logic -

**Pure Mathematics Dictionary**

 







 

 

 

 



 

 

 

 

Definition

A pair of results connecting differentiation and integration: (1) if f is integrable on [a,b] and F(x)=∫_a^x f(t) dt then F is continuous and F'(x)=f(x) almost everywhere under mild conditions; (2) if F is any antiderivative of f on [a,b] then ∫_a^b f(x) dx = F(b)-F(a).