Arithmetic Course/Number Operation/Integration/Indefinite Integration

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Indefinite Integration[edit]

Mathematic operation on a function to find the total area under the function's curve . Given a function of x f(x) then the Indefinite Integration of function f(x) has a symbol below

\int f(x) dx = Lim_{\Delta x \to 0} \Sigma \Delta x [f(x) + \frac{f(x + \Delta x)}{2}]

Result

\int_{ }^{ } f(x)\, dx = F(x) + C = \int f(x) dx = f^'(x) + C


Integration laws[edit]

\int { \frac { f^{'} (x)}  {   f (x)} } {\rm d}x = \ln | f (x) | + c
\int {UV} = U \int {V}  - \int {\left( U^{'} \int { V} \right) }
e^x also generates itself and is susceptible to the same treatment.
\int { e^{-x} \sin x }~ dx = ( -e^{-x} ) \sin x - \int { (-e^{-x}) \cos x} ~ dx
 =  -e^{-x}  \sin x + \int { e^{-x} \cos x } ~ dx
 =  -e^{-x}  (\sin x +  \cos x ) - \int { e^{-x} \sin x } ~ dx + c
We now have our required integral on both sides of the equation so
= - \frac 1 2 e^{-x} ( \sin x + \cos x ) + c


  • f (x) = m
\int m dx  = m x + C
  • f (x) = x^n
\int { f(x) }dx  = \frac {1}{n+1} x^{n+1} + c
  • f (x) = \frac{1}{x}
\int { \frac{1}{x}} dx = \ln x