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Integration Formulae

{Indefinite Integral}

Method of substitution

$\int f(g(x)) g^{\prime}(x) d x=\int f(u) d u$

Integration by parts

$\int f(x) g^{\prime}(x) d x=f(x) g(x)-\int g(x) f^{\prime}(x) d x$

Integrals of Rational and Irrational Functions

$\int x^{n} d x=\frac{x^{n+1}}{n+1}+C$

$\int \frac{1}{x} d x=\ln |x|+C$

$\int c d x=c x+C$

$\int x d x=\frac{x^{2}}{2}+C$

Limits and Differentiation Formulae

{1. Limits}

if $\lim _{x \rightarrow a} f(x)=l$ and $\lim _{x \rightarrow a} g(x)=m$, then

$\lim _{x \rightarrow a}[f(x) \pm g(x)]=l \pm m$

$\lim _{x \rightarrow a}[f(x) \cdot g(x)]=l \cdot m$

$\lim _{x \rightarrow a} \frac{f(x)}{g(x)}=\frac{l}{m}$ where $m \neq 0$

$\lim _{x \rightarrow a} c \cdot f(x)=c \cdot l$

$\lim _{x \rightarrow a} \frac{1}{f(x)}=\frac{1}{l}$ where $l \neq 0$

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