# Arithmetic Course/Number Operation/Integration/Indefinite Integration

## Indefinite Integration

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

$\int { \frac { f^{'} (x)} { f (x)} } {\rm d}x = \ln | f (x) | + c$
$\int {UV} = U \int {V} - \int {\left( U^{'} \int { V} \right) }$
$e^x$ 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
$= - \frac 1 2 e^{-x} ( \sin x + \cos x ) + c$

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