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L-Hospital’s rule

If at x = a, a function f(x) has an indeterminate value such as 529.pngetc. then L-hospital’s rule can be applied.

consider two functions f(x) and ϕ(x) having value 0 at x = 0, then 534.png

If still f ′ (x) and
ϕ′ (x) has value zero at x = a, then 539.png and goes on.

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  (c544.png

Standard Integration

  1. 549.png 
  2. 554.png559.png (a = constant)
  3. 564.pngwhen n = 1, i.e., 569.png 
  4. 574.png 
  5. 579.png 
  6. 584.png 
  7. 590.png 
  8. 595.png 
  9. 600.png 
  10. 605.png 
  11. 610.png  
  12. 615.png 
  13. 620.png
  14. 625.png 
  15. 630.png 
  16. 635.png 
  17. 641.png 
  18. 646.png 
  19. 651.png 
  20. 656.png 
  21. 661.png 
  22. 666.png 
  23. 671.png 
  24. 676.png 
  25. 681.png 
  26. 686.png 
  27. 692.png 

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 funtion, 
T- Trignomatric function, E-Exponential function.

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