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

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

• $f(x)=m$

$\int mdx=mx+C$

• $f(x)=x^{n}$

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

• $f(x)={\frac {1}{x}}$

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