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Mean Value Theorems

  1. Rolle’s theorem
     
    If a function f (x) is continuous in a closed interval [ab] i.e. 
     
    a ≤ x ≤ b, and derivable in open interval (ab) i.e. a < x < b, and if f (a) = f (b), then there exist atleast one real c in (ab) such that f ′(c) = 0
     
  2. Lagrange’s first mean value theorem
     
    If a function f (x) is continuous in closed interval [ab] and f’(x) exist in open interval (a, b) then there exist atleast one value of c such that,
     
    f  (cDescription: 468.png

Standard Integration

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  2. Description: 478.pngDescription: 483.png (a = constant)
  3. Description: 488.png  when n = 1, i.e., Description: 493.png
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    u is first function, v is second function. To find first function out of two we follow ‘ILATE’ rule.
     
    I-Inverse, L-Log function, A-Algebraic function, 
     
    T- Trigonometric function, E-Exponential function. 




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