                         INTEGRAL CALCULUS



The antiderivative of a function f(x), denoted by F(x), is defined as 

⌠                            , 
│ f(x)dx = F(x) + c  where  F (x) = f(x)  and c is an arbitrary constant.  
⌡  

The Fundamental Theorem of Calculus relates the antiderivative to the
integral as:

               b
              ⌠                               ,
              │ f(x)dx = F(b) - F(a)  where  F (x) = f(x)                 
              ⌡    
               a  

as well as:
                     x  
                  d ⌠
                 ── │ f(t)dt = f(x).   
                 dx ⌡ 
                     a


      If f(x) ≥ O and continuous on [a,b] then:
  b
⌠
│ f(x)dx  is the area under the curve y = f(x) bounded by the 
⌡ 
 a

lines x = a, x = b and y = O.


     Properties of integrals of functions f(x) and g(x) continuous
on [a,b]:
          a            
         ⌠
     1.  │ f(x)dx  =  O
         ⌡ 
          a

          b                b
         ⌠                ⌠
     2.  │ k f(x)dx  =  k │ f(x)dx       where k is a constant
         ⌡                ⌡
          a                a

          b                     b          b
         ⌠                     ⌠          ⌠   
     3.  │ (f(x) ± g(x))dx  =  │ f(x)dx ± │ g(x)dx
         ⌡                     ⌡          ⌡
          a                     a          a

          b            c            b
         ⌠            ⌠            ⌠   
     4.  │ f(x)dx  =  │ f(x)dx  +  │ f(x)dx    for  c in (a,b).
         ⌡            ⌡            ⌡
          a            a            c

          b             a   
         ⌠             ⌠
     5.  │ f(x)dx  =  -│ f(x)dx
         ⌡             ⌡
          a             b                                           

          b
         ⌠
     6.  │ f(x)dx  =  f(c)(b - a) for some  c  in [a,b].
         ⌡                        (Mean Value Theorem for Integrals)
          a   

          b                     g(b)
         ⌠          ,          ⌠                ,
     7.  │ f(g(x)) g (x)dx  =  │ f(u)du   when g (x) is continuous on
         ⌡                     ⌡          [a,b]. (u substitution)
          a                     g(a)
         
Notice that the limits of integration on the u variable correspond to
those on  x  through the function  u = g(x).

The concept of antidifferentiation leads us to the following formula:

         ⌠            n+1
         │  n        x
     8.  │ x dx  =  ──────  +  c
         │          n + 1
         ⌡


         ⌠
     9.  │ cos x dx  =  sin x  +  c
         ⌡



         ⌠
    1O.  │ sin x dx  =  -cos x  +  c
         ⌡

          
         ⌠    2
    11.  │ sec x dx  =  tan x  +  c
         ⌡

            
         ⌠    2
    12.  │ csc x dx  =  -cot x  +  c
         ⌡
                     
         ⌠
    13.  │ sec x tan x dx  =  sec x  +  c
         ⌡ 


         ⌠
    14.  │ csc x cot x dx  =  - csc x  +  c
         ⌡

where c is an arbitrary constant.
