Definition (Definite Integral): Let be continuous on the closed interval

Properties of the Riemann Integral (Continued)

Let be continuous on and appropriately differentiable. Then,

14. 
    Leibniz Integral Rule.
    

Proof: By the First FTIC and the Chain Rule, we have





14. 
    Let Then on Zero Rule.
    

Proof: The “if part” is trivial. Therefore, we shall only prove the “only if” part. So, assume that Since Now,

by continuity. 

14. 
    U-Substitution Rule.
    

Proof: Let Then, Thus,

. 

14. 
    Change of Variable Rule.
    

Proof: Let Then, 

14. 
    where Here is said to be even. A Symmetry Law.
    

14. 
    where Here is said to be odd. A Symmetry Law.
    
15. 
    Absolute Value Rule.
    

Proof: Clearly, we have on By the Comparison Rule, we then have By the Comparison Rule, we then have . 

14. 
    Definition of the average value of over
    
15. 
    First Mean Value Theorem for Integrals (First MVTI).
    

Proof: The theorem is trivial if is constant on Therefore, we may assume that is not constant on By the Max-Min Rule, we have that If then This would then imply that on This contradicts the fact that is not constant on Thus, In the same way, we also have Hence, Since is continuous on by the Intermediate Value Theorem, there exists such that 

14. 
    where is periodic with period Periodic Rule for Integrals.