# Luật toán tích phân bất định

${\displaystyle \int f(x)dx=\lim _{\Delta x\rightarrow 0}\sum [f(x)+{\frac {\Delta f(x)}{2}}]\Delta x=F(x)+C}$
 Rule Conditions 1 ${\displaystyle \int a\,dx=ax}$ 2 Homogeniety ${\displaystyle \int af(x)\,dx=a\int f(x)\,dx}$ 3 Associativity ${\displaystyle \int {\left(f\pm g\pm h\pm \cdots \right)\,dx}=\int f\,dx\pm \int g\,dx\pm \int h\,dx\pm \cdots }$ 4 Integration by Parts ${\displaystyle \int _{a}^{b}fg'\,dx=\left[fg\right]_{a}^{b}-\int _{a}^{b}gf'\,dx}$ 4 General Integration by Parts ${\displaystyle \int f^{(n)}g\,dx=f^{(n-1)}g'-f^{(n-2)}g''+\ldots +(-1)^{n}\int fg^{(n)}\,dx}$ 5 ${\displaystyle \int f(ax)\,dx={\frac {1}{a}}\int f(x)\,dx}$ 6 Substitution Rule ${\displaystyle \int g\{f(x)\}\,dx=\int g(u){\frac {dx}{du}}\,du=\int {\frac {g(u)}{f'(x)}}\,du}$ ${\displaystyle u=f(x)\,}$ 7 ${\displaystyle \int x^{n}\,dx={\frac {x^{n+1}}{n+1}}}$ ${\displaystyle n\neq -1\,}$ 8 ${\displaystyle \int {\frac {1}{x}}\,dx=\ln |x|}$ 9 ${\displaystyle \int e^{x}\,dx=e^{x}}$ 10 ${\displaystyle \int a^{x}\,dx={\frac {a^{x}}{\ln |a|}}}$ ${\displaystyle a\neq 1}$