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  1.  
  2.                            Review of Integration
  3.                            ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀
  4.     
  5.        List of standard integrals (constants of integration suppressed).
  6.  
  7.        
  8.                             ⌠  n              1     n + 1
  9.                             │ x   dx   =   ─────── x         ( n ╪ -1 )
  10.                             ⌡               n + 1
  11.  
  12.                            ⌠  1
  13.                            │ ───  dx   =   ln │x│
  14.                            ⌡  x
  15.  
  16.                            ⌠  x             x
  17.                            │ e   dx    =   e
  18.                            ⌡
  19.  
  20.                          ⌠
  21.                          │ sin x  dx   =   - cos x
  22.                          ⌡
  23.  
  24.                          ⌠
  25.                          │ cos x  dx   =   sin x
  26.                          ⌡
  27.  
  28.                          ⌠
  29.                          │ tan x  dx   =   ln │sec x│
  30.                          ⌡
  31.  
  32.                          ⌠
  33.                          │ cot x  dx   =   ln │sin x│
  34.                          ⌡
  35.  
  36.                          ⌠
  37.                          │ sec x  dx   =   ln │sec x + tan x│
  38.                          ⌡
  39.  
  40.                          ⌠
  41.                          │ csc x  dx   =   ln │csc x - cot x│
  42.                          ⌡
  43.  
  44.                          ⌠    2
  45.                          │ sec x  dx   =   tan x
  46.                          ⌡
  47.  
  48.                    ⌠
  49.                    │ sec x tan x  dx   =   sec x
  50.                    ⌡
  51.  
  52.                       ⌠     1                 -1
  53.                       │ ────────  dx   =   tan  x
  54.                       ⌡       2
  55.                          1 + x  
  56.  
  57.                    ⌠      1                  -1
  58.                    │ ───────────  dx   =   sin  x
  59.                    ⌡       2 1/2
  60.                      (1 - x ) 
  61.  
  62.  
  63.  
  64.        List of standard theorems
  65.  
  66.                         ⌠  df(x)
  67.                         │ ──────  dx   =   f(x)
  68.                         ⌡   dx
  69.  
  70.                     ⌠                      ⌠             ⌠
  71.                     │ f(x) ± g(x) dx   =   │ f(x) dx  ±  │ g(x) dx
  72.                     ⌡                      ⌡             ⌡
  73.  
  74.                          ⌠                   ⌠
  75.                          │ k f(x) dx   =   k │ f(x) dx     (k constant)
  76.                          ⌡                   ⌡
  77.  
  78.                          ⌠                   ⌠
  79.                          │ u dv   =   uv  -  │ v du    (integration by parts)
  80.                          ⌡                   ⌡
  81.  
  82.                      ⌠       du            ⌠
  83.                      │ f(u) ────  dx   =   │ f(u) du   (u - substitution)
  84.                      ⌡       dx            ⌡
  85.  
  86.  
  87.  
  88. ┌─────────────────┐
  89. │ Common Mistakes │
  90. └─────────────────┘
  91.  
  92.  
  93.                     ⌠                    ⌠          ⌠
  94. WRONG               │ f(x) g(x) dx   =   │ f(x) dx  │ g(x) dx
  95.                     ⌡                    ⌡          ⌡
  96.  
  97.                                               ⌠
  98. There is NO general formula which simplifies  │ f(x) g(x) dx.  The nearest
  99.                                               ⌡
  100. result for integrating a product is integration by parts.
  101.  
  102.                   ⌠                      ⌠           ⌠
  103. WRONG             │ f(x) ÷ g(x) dx   =   │ f(x) dx ÷ │ g(x) dx
  104.                   ⌡                      ⌡           ⌡
  105.  
  106.                                               ⌠
  107. There is NO general formula which simplifies  │ f(x) ÷ g(x) dx.
  108.                                               ⌡
  109.  
  110.  
  111. This is the end of the help file.  Press the ESC key to return to the quiz
  112. question you were doing.
  113.