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

$\int f(x)dx=Li{m}_{\mathrm{\Delta }x\to 0}\mathrm{\Sigma }\mathrm{\Delta }x[f(x)+\frac{f(x+\mathrm{\Delta }x)}{2}]$

Result

$\underset{}{\overset{}{\int }}f(x)dx=F(x)+C=\int f(x)dx=f^{\text{'}}(x)+C$

Integration laws

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

$\int UV=U\int V-\int (U^{\text{'}}\int V)$

$e^x$ also generates itself and is susceptible to the same treatment.

$\int e^{-x}\sin xdx=(-e^{-x})\sin x-\int (-e^{-x})\cos xdx$

$=-e^{-x}\sin x+\int e^{-x}\cos xdx$

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

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

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

• $f(x)=m$

$\int mdx=mx+C$

• $f(x)=x^n$

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

• $f(x)=\frac{1}{x}$

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