Chit-fûn-péu

Chit-fûn-péu

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• Fán sûng-chì hàm-su chit-fûn-péu

Pâu-hàm $\sqrt{a+bx}$ ke chit-fûn-péu

$\int \frac{1}{x\sqrt{a+bx}}dx=\frac{1}{\sqrt{a}}\ln \left(\frac{\sqrt{a+bx}-\sqrt{a}}{\sqrt{a+bx}+\sqrt{a}}\right)+C,a>0$

Pâu-hàm $R=\sqrt{|a|x^2+bx+c}(a\ne 0)$ ke chit-fûn-péu

$\int \frac{dx}{R}=\frac{1}{\sqrt{a}}\ln (2\sqrt{a}R+2ax+b)(\text{for }a>0)$

$\int \frac{dx}{R}=\frac{1}{\sqrt{a}}\mathrm{arsinh}\frac{2ax+b}{\sqrt{4ac-b^2}}\text{(for }a>0\text{, }4ac-b^2>0\text{)}$

$\int \frac{dx}{R}=\frac{1}{\sqrt{a}}\ln |2ax+b|\text{(for }a>0\text{, }4ac-b^2=0\text{)}$

$\int \frac{dx}{R}=-\frac{1}{\sqrt{-a}}\arcsin \frac{2ax+b}{\sqrt{b^2-4ac}}\text{(for }a<0\text{, }4ac-b^2<0\text{, }(2ax+b)<\sqrt{b^2-4ac}\text{)}$

$\int \frac{dx}{R^3}=\frac{4ax+2b}{(4ac-b^2)R}$

$\int \frac{dx}{R^5}=\frac{4ax+2b}{3(4ac-b^2)R}(\frac{1}{R^2}+\frac{8a}{4ac-b^2})$

$\int \frac{dx}{R^{2n+1}}=\frac{2}{(2n-1)(4ac-b^2)}[\frac{2ax+b}{R^{2n-1}}+4a(n-1)\int \frac{dx}{R^{2n-1}}]$

$\int \frac{x}{R}dx=\frac{R}{a}-\frac{b}{2a}\int \frac{dx}{R}$

$\int \frac{x}{R^3}dx=-\frac{2bx+4c}{(4ac-b^2)R}$

$\int \frac{x}{R^{2n+1}}dx=-\frac{1}{(2n-1)a{R}^{2n-1}}-\frac{b}{2a}\int \frac{dx}{R^{2n+1}}$

$\int \frac{dx}{xR}=-\frac{1}{\sqrt{c}}\ln \left(\frac{2\sqrt{c}R+bx+2c}{x}\right)$

$\int \frac{dx}{xR}=-\frac{1}{\sqrt{c}}\mathrm{arsinh}\left(\frac{bx+2c}{|x|\sqrt{4ac-b^2}}\right)$

Thin-chit-fûn

$\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\text{sin}}^{n}x\text{d}x=\underset{0}{\overset{\frac{\pi }{2}}{\int }}{\text{cos}}^{n}x\text{d}x=\{\begin{array}{cc}\frac{n-1}{n}\cdot \frac{n-3}{n-2}\cdot \dots \cdot \frac{4}{5}\cdot \frac{2}{3},& \text{if }n>1\text{ and }n\text{ is odd}\\ \frac{n-1}{n}\cdot \frac{n-3}{n-2}\cdot \dots \cdot \frac{3}{4}\cdot \frac{1}{2}\cdot \frac{\pi }{2},& \text{if }n>0\text{ and }n\text{ is even}\end{array}$