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Kien-li̍p sû-sitHâ-chai vì PDFCho-tet yin-chho ke pán-pún 外觀 移至側邊欄 隱藏 Chhiùng Wikipedia lòi Chit-fûn-péu Yù-lî hàm-su chit-fûn-péu Mò-lî hàm-su chit-fûn-péu Sâm-kok hàm-su chit-fûn-péu Chṳ́-su hàm-su chit-fûn-péu Tui-su hàm-su chit-fûn-péu Fán sâm-kok hàm-su chit-fûn-péu Sûng-chì hàm-su chit-fûn-péu Fán sûng-chì hàm-su chit-fûn-péu Pâu-hàm a x + b {\displaystyle ax+b} ke chit-fûn-péu[phiên-siá | kói ngièn-sṳ́-mâ] Pâu-hàm a + b x {\displaystyle {\sqrt {a+bx}}} ke chit-fûn-péu[phiên-siá | kói ngièn-sṳ́-mâ] ∫ 1 x a + b x d x = 1 a ln ( a + b x − a a + b x + a ) + C , a > 0 {\displaystyle \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 x 2 ± α 2 {\displaystyle x^{2}\pm \alpha ^{2}} ke chit-fûn-péu[phiên-siá | kói ngièn-sṳ́-mâ] Pâu-hàm a x 2 + b {\displaystyle {ax^{2}+b}} ke chit-fûn-péu[phiên-siá | kói ngièn-sṳ́-mâ] Pâu-hàm a x 2 + b x + c ( a > 0 ) {\displaystyle ax^{2}+bx+c\qquad (a>0)} ke chit-fûn-péu[phiên-siá | kói ngièn-sṳ́-mâ] Pâu-hàm a 2 + x 2 ( a > 0 ) {\displaystyle {\sqrt {a^{2}+x^{2}}}\qquad (a>0)} ke chit-fûn-péu[phiên-siá | kói ngièn-sṳ́-mâ] Pâu-hàm x 2 − a 2 ( x 2 > a 2 ) {\displaystyle {\sqrt {x^{2}-a^{2}}}\qquad {(x^{2}>a^{2})}} ke chit-fûn-péu[phiên-siá | kói ngièn-sṳ́-mâ] Pâu-hàm a 2 − x 2 ( a 2 > x 2 ) {\displaystyle {\sqrt {a^{2}-x^{2}}}\qquad (a^{2}>x^{2})} ke chit-fûn-péu[phiên-siá | kói ngièn-sṳ́-mâ] Pâu-hàm R = | a | x 2 + b x + c ( a ≠ 0 ) {\displaystyle R={\sqrt {|a|x^{2}+bx+c}}\qquad (a\neq 0)} ke chit-fûn-péu[phiên-siá | kói ngièn-sṳ́-mâ] ∫ d x R = 1 a ln ( 2 a R + 2 a x + b ) ( for a > 0 ) {\displaystyle \int {\frac {dx}{R}}={\frac {1}{\sqrt {a}}}\ln \left(2{\sqrt {a}}R+2ax+b\right)\qquad ({\mbox{for }}a>0)} ∫ d x R = 1 a arsinh 2 a x + b 4 a c − b 2 (for a > 0 , 4 a c − b 2 > 0 ) {\displaystyle \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{)}}} ∫ d x R = 1 a ln | 2 a x + b | (for a > 0 , 4 a c − b 2 = 0 ) {\displaystyle \int {\frac {dx}{R}}={\frac {1}{\sqrt {a}}}\ln |2ax+b|\quad {\mbox{(for }}a>0{\mbox{, }}4ac-b^{2}=0{\mbox{)}}} ∫ d x R = − 1 − a arcsin 2 a x + b b 2 − 4 a c (for a < 0 , 4 a c − b 2 < 0 , ( 2 a x + b ) < b 2 − 4 a c ) {\displaystyle \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{)}}} ∫ d x R 3 = 4 a x + 2 b ( 4 a c − b 2 ) R {\displaystyle \int {\frac {dx}{R^{3}}}={\frac {4ax+2b}{(4ac-b^{2})R}}} ∫ d x R 5 = 4 a x + 2 b 3 ( 4 a c − b 2 ) R ( 1 R 2 + 8 a 4 a c − b 2 ) {\displaystyle \int {\frac {dx}{R^{5}}}={\frac {4ax+2b}{3(4ac-b^{2})R}}\left({\frac {1}{R^{2}}}+{\frac {8a}{4ac-b^{2}}}\right)} ∫ d x R 2 n + 1 = 2 ( 2 n − 1 ) ( 4 a c − b 2 ) [ 2 a x + b R 2 n − 1 + 4 a ( n − 1 ) ∫ d x R 2 n − 1 ] {\displaystyle \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]} ∫ x R d x = R a − b 2 a ∫ d x R {\displaystyle \int {\frac {x}{R}}\;dx={\frac {R}{a}}-{\frac {b}{2a}}\int {\frac {dx}{R}}} ∫ x R 3 d x = − 2 b x + 4 c ( 4 a c − b 2 ) R {\displaystyle \int {\frac {x}{R^{3}}}\;dx=-{\frac {2bx+4c}{(4ac-b^{2})R}}} ∫ x R 2 n + 1 d x = − 1 ( 2 n − 1 ) a R 2 n − 1 − b 2 a ∫ d x R 2 n + 1 {\displaystyle \int {\frac {x}{R^{2n+1}}}\;dx=-{\frac {1}{(2n-1)aR^{2n-1}}}-{\frac {b}{2a}}\int {\frac {dx}{R^{2n+1}}}} ∫ d x x R = − 1 c ln ( 2 c R + b x + 2 c x ) {\displaystyle \int {\frac {dx}{xR}}=-{\frac {1}{\sqrt {c}}}\ln \left({\frac {2{\sqrt {c}}R+bx+2c}{x}}\right)} ∫ d x x R = − 1 c arsinh ( b x + 2 c | x | 4 a c − b 2 ) {\displaystyle \int {\frac {dx}{xR}}=-{\frac {1}{\sqrt {c}}}\operatorname {arsinh} \left({\frac {bx+2c}{|x|{\sqrt {4ac-b^{2}}}}}\right)} Pâu-hàm sâm-kok hàm-su ke chit-fûn-péu[phiên-siá | kói ngièn-sṳ́-mâ] Pâu-hàm fán sâm-kok hàm-su ke chit-fûn-péu[phiên-siá | kói ngièn-sṳ́-mâ] Pâu-hàm chṳ́-su hàm-su ke chit-fûn-péu[phiên-siá | kói ngièn-sṳ́-mâ] Pâu-hàm tui-su hàm-su ke chit-fûn-péu[phiên-siá | kói ngièn-sṳ́-mâ] Pâu-hàm Sûng-chì hàm-su ke chit-fûn-péu[phiên-siá | kói ngièn-sṳ́-mâ] Thin-chit-fûn[phiên-siá | kói ngièn-sṳ́-mâ] ∫ 0 π 2 sin n x d x = ∫ 0 π 2 cos n x d x = { n − 1 n ⋅ n − 3 n − 2 ⋅ … ⋅ 4 5 ⋅ 2 3 , if n > 1 and n is odd n − 1 n ⋅ n − 3 n − 2 ⋅ … ⋅ 3 4 ⋅ 1 2 ⋅ π 2 , if n > 0 and n is even {\displaystyle \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}}}
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