# Calculus Course/Integration/Indefinite Integral

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## Indefinite Integral

Indefinite Integral is a mathematic operation on a non linear function over an indefinite a boundary

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

## Indefinite Integral Rules

Because antidifferentiation is the inverse operation of the differentiation, antidifferentiation theorems and rules are obtained from those on differentiation. Thus, the following theorems can be proven from the corresponding differentiation theorems:

• General antidifferentiation rule:
${\displaystyle \int dx=x+C}$
• The general antiderivative of a constant times a function is the constant multiplied by the general antiderivative of the function:
${\displaystyle \int af(x)\,dx=a\int f(x)\,dx+C}$
• If ƒ and g are defined on the same interval, then the general antiderivative of the sum of ƒ and g equals the sum of the general antiderivatives of ƒ and g:
${\displaystyle \int [f(x)+g(x)]\,dx=\int f(x)\,dx+\int g(x)\,dx+C}$
• If n is a real number,
${\displaystyle \int x^{n}\,dx={\begin{cases}{\frac {x^{n+1}}{n+1}}+C,&{\text{if }}n\neq -1\\[6pt]\ln |x|+C,&{\text{if }}n=-1\end{cases}}}$

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

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