INTEGRAL CALCULUS
                      Constant of Integration: 
                        dd
                                                                             . 
                                                      
                            F x f x                  F x c  f x
                                                                  
                       dx                         dx 
                      Therefore, f(x) dx = F(x) + c. 
                      Properties of Indefinite Integration: 
                      (i)        af(x)dx  a f(x)dx.
                               
                      (ii)        f(x)  g(x) dx         f(x)dx      g(x)dx.
                                               
                                                                   
                      (iii) If  f(u)du = F(u) + c,                                     then  f(ax + b) dx = 1F axb c, a0.
                                                                                                                                    a            
                      Integration as the Inverse Process of Differentiation 
                      Basic formulae: 
                      Antiderivatives or integrals of some of the widely used functions 
                      (integrands) are given below. 
                                         n1                                                            n 1
                                      
                                   d x                                                                  x
                                                     nn
                                                x                                      x dx                c , n  – 1
                                      
                                                                                      
                                  dx n1                                                               n 1
                                      
                                   d                 1                                  1
                                     (ln | x | )                                        dx  ln| x | c
                                  dx                 x                                 x
                                   d     x       x                                       x            x
                                     (e )  e                                        e dx  e c
                                  dx                                                 
                                                                                                         x
                                   da
                                     (ax)  (ax lna)                                   ax dx              c         ( a> 0)
                                  dx                                                                   lna
                                   d
                                     (sinx)  cos x                                   cos x dx  sinx c
                                  dx                                                  
                                   d
                                     (cos x)  sinx                                  sin x dx  cos x c
                                  dx                                                  
                                   d                   22
                                     ( tanx)  sec       x                             sec x dx  tan x c
                                  dx                                                   
                                                                                                                                                                  1 
                        
                                    d (cosec x)  (cotx cosec x)                                      cosecx cotx dx  cosecx c  
                                    dx                                                                 
                                    d (secx)  sec x tanx                                             secx tanx dx  secx c  
                                    dx                                                                
                                     d                           22
                                       (cotx)  cosec x                                    cosec x dx  cot x  c  
                                    dx                                                      
                                     d          x             1                                   1                           x
                                                                                                                            
                                             11
                                       (sin      )                                                      dx  sin               c 
                                                                                                                            
                                    dx          a            2     2                            2      2                    a
                                                                                                                            
                                                           a x                               a      x
                                     d           x          a                                    dx          1          x
                                             11
                                       (tan       )                                                        tan         c 
                                    dx           a        2     2                           2        2     a           a
                                                        x a                                 x      a
                                     d                          1                                   1
                                              11
                                       (sec     x)                                                       dx  sec (x) c  
                                                                                           
                                    dx                             22
                                                         | x |   x 1                        x    x      1
                                     cotx dx               cosx dx         ln|sinx|  c 
                                    
                                                             sinx
                                     tanx dx           sinx dx  ln|cosx|c or ln|secx|c 
                                    
                                                          cosx
                                                         secx(secx tanx)                                                                 x     
                                                                                                                                         
                                     secx dx                                      dx  ln|secx  tanx| c or ln tan                                c  
                                                                                                                                         
                                    
                                                             secx  tanx                                                                  2     4
                                                                                                                                         
                                     cosecxdx  cosecx(cotx cosecx)dx ln (cot x  cosec x)| c or ln tan x  c  
                                    
                                                                 cotx cosecx                                                                     2
                        
                       Standard Formulae: 
                                          dx                      22 
                                                    ln x       x a c
                                         22
                                        xa
                                          dx                      22 
                                                    ln x       x a c
                                         22
                                        xa
                                         dx          1      x  a
                                                  ln                 c
                                                                          
                                     22
                                      xa            2a     x  a
                                         dx          1      ax
                                                  ln                 c
                                                                          
                                     22
                                      ax            2a     ax
                                                                              2
                                                          ua
                                       u2  a2du            u2 a2           ln u      u2 a2 c 
                                                         22
                                                                              2
                                                         ua
                                       u2 a2du             u2 a2           ln u      u2 a2 c  
                                                         22
                                         2     2          x      2     2    a2        -1 x           
                                        a x  dx =              a x              sin         c
                                                          2                  2           a
                        
                                                                                                                                                                           2 
                        
                    
                   Integration by Substitution: 
                   There are following types of substitutions.  
                   Direct Substitution:  
                   If  integral is of the form  f(g(x)) g(x) dx, then put g(x) = t, 
                   provided  f(t) dt exists. 
                                      
                   Standard Substitutions: 
                                                                        2        2          22
                                                                                           xa
                            For terms of the form x  + a  or                                       , put x = a tan  or  a 
                             cot 
                                                                        2       2          22
                                                                                          xa
                            For terms of the form x  - a  or                                      , put x = a sec   or  a 
                             cosec 
                                                                        2       2          22
                                                                                          ax
                            For terms of the form a  - x  or                                      , put x = a sin   or  a 
                             cos 
                                            ax        ax
                            If both               ,          are present, then put x = a cos. 
                                                                                                 2                 2
                            For the type                              , put x = a cos  + b sin  
                                                        xa b x
                                                                  
                                                                     nn
                            For the type  x2 a2 x or  x x2 a2 , put the expression within 
                                                                                       
                             the bracket = t. 
                                                                1111
                                                             11                     x b n        1
                                                                                      
                            For the type                       nn  (n N, n >1),put
                                                      x a        x   b         or
                                                                     
                                                                                       2
                                                                                        xa
                                                                                      
                                                                                                    xa
                                                                                                        
                             xb       . 
                                     t
                             xa
                            For              1           , n ,n  N  (and > 1), again put (x + a) = t (x 
                                                              1    2
                                            nn
                                             12
                                      x a     x   b
                                                   
                             + b) 
                    
                    
                    
                                                                                                                                           3 
                    
                                            
                                           Integration by Parts: 
                                           If u and v be two functions of x, then integral of product of these 
                                                                                                                                                                                                                   du
                                                                                                                                                                                                                 
                                           two functions is given by:  uv dx u v dx-                                                                                                                                          v dx dx 
                                                                                                                                                                                                                 
                                                                                                                                                                                                           dx
                                                                                                                                                                                                                 
                                           (Inverse, Logarithmic, Algebraic, Trigonometric, 
                                           Exponential) 
                                           In the above stated order, the function on the left is always chosen 
                                           as the first function. This rule is called as ILATE e.g. In the 
                                           integration of xsinxdx, x is taken as the first function and sinx is 
                                                                                                       
                                           taken as the second function. 
                                           An important result: In the integral g(x)exdx,if g(x) can be 
                                                                                                                                                                                                     
                                                                                                                                                                                                             x                                                      x 
                                           expressed as g(x) = f(x) + f(x) then                                                                                                                                                                      = e f(x) + c 
                                                                                                                                                                                                         e f(x)f (x)dx
                                                                                                                                                                                                                                             
                                                                                                                                                                                                     
                                            
                                           Integration By Partial Fractions: 
                                           A  function  of  the  form  P(x)/Q(x),  where  P(x)  and  Q(x)  are 
                                           polynomials, is called a rational function. Consider the rational 
                                           function                                              x  7                                   1           -           1            
                                                                                   (2x - 3) (3x + 4)                               2x 3                    3x + 4
                                                                                                     k...               2                                               r
                                           Q(x) = (x - a)                                                      (x  + x + )  ... where binomials are different, 
                                           and then set 
                                             P(x)                  A                      A                                     A                      Mx + N                                M x + N                                                Mx + N
                                                          =            1        +              2         + ... +                    k                      1               1  +                  2                2         ...                       r               r        ... 
                                             Q(x)                 (x-a)                            2                                    k                2                                    2                         2                            2                        r
                                                                                       (x-a)                                (x-a)                    x x (x x)                                                                         (x x)
                                           Algorithm to express the infinite series as definite integral: 
                                                                                                                                                                                                                                         1               r
                                                                                                                                                                                                                                               f  
                                           (i) Express the given series in the form of                                                                                                                                                                            
                                                                                                                                                                                                                                  
                                                                                                                                                                                                                                         nn
                                                                                                                                                                                                                                                     
                                                                                                                                                                                                                  n1 1                    r
                                                                                                                                                                                                        lim                   .f      
                                           (ii) The limit when n  is its sum                                                                                                                                                                      
                                                                                                                                                                                                         h0                          
                                                                                                                                                                                                                          nn
                                                                                                                                                                                                                  r0                  
                                                                                                                                                                                                                                                                                                                              4