
                           Review of Integration
                           ▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀▀
    
       List of standard integrals (constants of integration suppressed).

       
                            ⌠  n              1     n + 1
                            │ x   dx   =   ─────── x         ( n ╪ -1 )
                            ⌡               n + 1

                           ⌠  1
                           │ ───  dx   =   ln │x│
                           ⌡  x

                           ⌠  x             x
                           │ e   dx    =   e
                           ⌡

                         ⌠
                         │ sin x  dx   =   - cos x
                         ⌡

                         ⌠
                         │ cos x  dx   =   sin x
                         ⌡

                         ⌠
                         │ tan x  dx   =   ln │sec x│
                         ⌡

                         ⌠
                         │ cot x  dx   =   ln │sin x│
                         ⌡

                         ⌠
                         │ sec x  dx   =   ln │sec x + tan x│
                         ⌡

                         ⌠
                         │ csc x  dx   =   ln │csc x - cot x│
                         ⌡

                         ⌠    2
                         │ sec x  dx   =   tan x
                         ⌡

                   ⌠
                   │ sec x tan x  dx   =   sec x
                   ⌡

                      ⌠     1                 -1
                      │ ────────  dx   =   tan  x
                      ⌡       2
                         1 + x  

                   ⌠      1                  -1
                   │ ───────────  dx   =   sin  x
                   ⌡       2 1/2
                     (1 - x ) 



       List of standard theorems

                        ⌠  df(x)
                        │ ──────  dx   =   f(x)
                        ⌡   dx

                    ⌠                      ⌠             ⌠
                    │ f(x) ± g(x) dx   =   │ f(x) dx  ±  │ g(x) dx
                    ⌡                      ⌡             ⌡

                         ⌠                   ⌠
                         │ k f(x) dx   =   k │ f(x) dx     (k constant)
                         ⌡                   ⌡

                         ⌠                   ⌠
                         │ u dv   =   uv  -  │ v du    (integration by parts)
                         ⌡                   ⌡

                     ⌠       du            ⌠
                     │ f(u) ────  dx   =   │ f(u) du   (u - substitution)
                     ⌡       dx            ⌡



┌─────────────────┐
│ Common Mistakes │
└─────────────────┘


                    ⌠                    ⌠          ⌠
WRONG               │ f(x) g(x) dx   =   │ f(x) dx  │ g(x) dx
                    ⌡                    ⌡          ⌡

                                              ⌠
There is NO general formula which simplifies  │ f(x) g(x) dx.  The nearest
                                              ⌡
result for integrating a product is integration by parts.

                  ⌠                      ⌠           ⌠
WRONG             │ f(x) ÷ g(x) dx   =   │ f(x) dx ÷ │ g(x) dx
                  ⌡                      ⌡           ⌡

                                              ⌠
There is NO general formula which simplifies  │ f(x) ÷ g(x) dx.
                                              ⌡


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