Basic

• Definition of derivative

  \[f' = \frac{d}{dx}f(x) = \lim_{h \to 0}\frac{f(x+h) - f(x)}{h}\] \[\frac{d}{dx}cx^n = cn x^{n-1}\]

• Definition of Indefinite Integral

  \[\int f(x) dx\] \[\frac{d}{dx}f(x) = F(x), \int F(x) dx = f(x) + c\] \[\int x^n dx = (\frac{1}{n+1}) x^{n+1} + c\]

• Integration by Parts 分部積分法

  \[\int u dv = uv - \int v du\]

Properties of Indefinite Integral

\[\int 2 f(x) dx = 2 \int f(x) dx\] \[\int[f(x) + g(x)] dx = \int f(x) dx + \int g(x) dx\] \[\int (x^3+x^2+x) dx = \int x^3 dx + \int x^2 dx + \int x dx = \frac{x^4}{4} + \frac{x^3}{3} + \frac{x^2}{2} + C\]

Common indefinite integrals

\[\int \frac{1}{x} dx = \ln{|x|} + c\] \[\int 0 dx = c\] \[\int 2 dx = 2x + c\] \[\int x^a dx = \frac{x{a+1}}{a+1} + c, a \neq -1\] \[\int e^x dx = e^x + c\] \[\int a^x dx = \frac{a^x}{\ln{a}} + c\] \[\int \cos x dx = \sin x + c\] \[\int \sin x dx = - \cos x + c\] \[\int \tan x dx = \ln | \sec x | + c\] \[\int \sec^2x\,dx = \tan x + c\] \[\int \cot x dx = \ln | \sin x | + c\] \[\int \big[ u(x) + v(x) \big] dx = \int u(x)\,dx + \int v(x)\,dx\]

Definite integrals

\[\int_{a}^{b} f'(x) dx = f(b) - f(a)\] \[F(x) \big|_{a}^{b} = F(b) - F(a)\] \[\int_{a}^{b} x^n\,dx = (\frac{1}{n+1}) (x^{n+1}) \big|_a^b = (\frac{1}{n+1})(b^{n+1} - a^{n+1})\] \[\int_a^b Cf(x)\,dx = C \int_a^b f(x)\,dx\] \[\int_a^b [f(x) + g(x)]\,dx = \int_a^b f(x)\,dx + \int_a^b g(x)\,dx\]