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
 * $$= - \frac 1 2 e^{-x} ( \sin x + \cos x ) + c$$


 * $$f (x) = m$$
 * $$\int m dx = m x + C$$


 * $$f (x) = x^n$$
 * $$\int { f(x) }dx = \frac {1}{n+1} x^{n+1} + c$$


 * $$f (x) = \frac{1}{x}$$
 * $$\int { \frac{1}{x}} dx = \ln x$$