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

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

Result

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

Integration laws

${\displaystyle \int {\frac {f^{'}(x)}{f(x)}}{\rm {d}}x=\ln |f(x)|+c}$

${\displaystyle \int {UV}=U\int {V}-\int {\left(U^{'}\int {V}\right)}}$

${\displaystyle e^{x}}$ also generates itself and is susceptible to the same treatment.

${\displaystyle \int {e^{-x}\sin x}~dx=(-e^{-x})\sin x-\int {(-e^{-x})\cos x}~dx}$

${\displaystyle =-e^{-x}\sin x+\int {e^{-x}\cos x}~dx}$

${\displaystyle =-e^{-x}(\sin x+\cos x)-\int {e^{-x}\sin x}~dx+c}$

We now have our required integral on both sides of the equation so

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

• ${\displaystyle f(x)=m}$

${\displaystyle \int mdx=mx+C}$

• ${\displaystyle f(x)=x^{n}}$

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

• ${\displaystyle f(x)={\frac {1}{x}}}$

${\displaystyle \int {\frac {1}{x}}dx=\ln x}$