Chit-fûn-péu

Chit-fûn-péu

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

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

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

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

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

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

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

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

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

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

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

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

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

Thin-chit-fûn

$\int _{0}^{\frac {\pi }{2}}{\mbox{sin}}^{n}x{\mbox{d}}x=\int _{0}^{\frac {\pi }{2}}{\mbox{cos}}^{n}x{\mbox{d}}x={\begin{cases}{\frac {n-1}{n}}\cdot {\frac {n-3}{n-2}}\cdot \ldots \cdot {\frac {4}{5}}\cdot {\frac {2}{3}},&{\mbox{if }}n>1{\mbox{ and }}n{\mbox{ is odd}}\\{\frac {n-1}{n}}\cdot {\frac {n-3}{n-2}}\cdot \ldots \cdot {\frac {3}{4}}\cdot {\frac {1}{2}}\cdot {\frac {\pi }{2}},&{\mbox{if }}n>0{\mbox{ and }}n{\mbox{ is even}}\end{cases}}$