Seznam integrálů iracionálních funkcí - List of integrals of irrational functions
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Následuje seznam integrály (primitivní funkce) z iracionální funkce . Úplný seznam integrálních funkcí najdete v části seznamy integrálů . V celém tomto článku konstanta integrace je pro stručnost vynechán.
Integrály zahrnující r = √A 2 + X 2 ∫ r d X = 1 2 ( X r + A 2 ln ( X + r ) ) { displaystyle int r , dx = { frac {1} {2}} left (xr + a ^ {2} , ln left (x + r right) right)} ∫ r 3 d X = 1 4 X r 3 + 3 8 A 2 X r + 3 8 A 4 ln ( X + r ) { displaystyle int r ^ {3} , dx = { frac {1} {4}} xr ^ {3} + { frac {3} {8}} a ^ {2} xr + { frac { 3} {8}} a ^ {4} ln left (x + r right)} ∫ r 5 d X = 1 6 X r 5 + 5 24 A 2 X r 3 + 5 16 A 4 X r + 5 16 A 6 ln ( X + r ) { displaystyle int r ^ {5} , dx = { frac {1} {6}} xr ^ {5} + { frac {5} {24}} a ^ {2} xr ^ {3} + { frac {5} {16}} a ^ {4} xr + { frac {5} {16}} a ^ {6} ln left (x + r right)} ∫ X r d X = r 3 3 { displaystyle int xr , dx = { frac {r ^ {3}} {3}}} ∫ X r 3 d X = r 5 5 { displaystyle int xr ^ {3} , dx = { frac {r ^ {5}} {5}}} ∫ X r 2 n + 1 d X = r 2 n + 3 2 n + 3 { displaystyle int xr ^ {2n + 1} , dx = { frac {r ^ {2n + 3}} {2n + 3}}} ∫ X 2 r d X = X 3 r 4 + A 2 X r 8 − A 4 8 ln ( X + r ) { displaystyle int x ^ {2} r , dx = { frac {x ^ {3} r} {4}} + { frac {a ^ {2} xr} {8}} - { frac {a ^ {4}} {8}} ln left (x + r right)} ∫ X 2 r 3 d X = X r 5 6 − A 2 X r 3 24 − A 4 X r 16 − A 6 16 ln ( X + r ) { displaystyle int x ^ {2} r ^ {3} , dx = { frac {xr ^ {5}} {6}} - { frac {a ^ {2} xr ^ {3}} { 24}} - { frac {a ^ {4} xr} {16}} - { frac {a ^ {6}} {16}} ln left (x + r right)} ∫ X 3 r d X = r 5 5 − A 2 r 3 3 { displaystyle int x ^ {3} r , dx = { frac {r ^ {5}} {5}} - { frac {a ^ {2} r ^ {3}} {3}}} ∫ X 3 r 3 d X = r 7 7 − A 2 r 5 5 { displaystyle int x ^ {3} r ^ {3} , dx = { frac {r ^ {7}} {7}} - { frac {a ^ {2} r ^ {5}} { 5}}} ∫ X 3 r 2 n + 1 d X = r 2 n + 5 2 n + 5 − A 2 r 2 n + 3 2 n + 3 { displaystyle int x ^ {3} r ^ {2n + 1} , dx = { frac {r ^ {2n + 5}} {2n + 5}} - { frac {a ^ {2} r ^ {2n + 3}} {2n + 3}}} ∫ X 4 r d X = X 3 r 3 6 − A 2 X r 3 8 + A 4 X r 16 + A 6 16 ln ( X + r ) { displaystyle int x ^ {4} r , dx = { frac {x ^ {3} r ^ {3}} {6}} - { frac {a ^ {2} xr ^ {3}} {8}} + { frac {a ^ {4} xr} {16}} + { frac {a ^ {6}} {16}} ln left (x + r right)} ∫ X 4 r 3 d X = X 3 r 5 8 − A 2 X r 5 16 + A 4 X r 3 64 + 3 A 6 X r 128 + 3 A 8 128 ln ( X + r ) { displaystyle int x ^ {4} r ^ {3} , dx = { frac {x ^ {3} r ^ {5}} {8}} - { frac {a ^ {2} xr ^ {5}} {16}} + { frac {a ^ {4} xr ^ {3}} {64}} + { frac {3a ^ {6} xr} {128}} + { frac {3a ^ {8}} {128}} ln vlevo (x + r doprava)} ∫ X 5 r d X = r 7 7 − 2 A 2 r 5 5 + A 4 r 3 3 { displaystyle int x ^ {5} r , dx = { frac {r ^ {7}} {7}} - { frac {2a ^ {2} r ^ {5}} {5}} + { frac {a ^ {4} r ^ {3}} {3}}} ∫ X 5 r 3 d X = r 9 9 − 2 A 2 r 7 7 + A 4 r 5 5 { displaystyle int x ^ {5} r ^ {3} , dx = { frac {r ^ {9}} {9}} - { frac {2a ^ {2} r ^ {7}} { 7}} + { frac {a ^ {4} r ^ {5}} {5}}} ∫ X 5 r 2 n + 1 d X = r 2 n + 7 2 n + 7 − 2 A 2 r 2 n + 5 2 n + 5 + A 4 r 2 n + 3 2 n + 3 { displaystyle int x ^ {5} r ^ {2n + 1} , dx = { frac {r ^ {2n + 7}} {2n + 7}} - { frac {2a ^ {2} r ^ {2n + 5}} {2n + 5}} + { frac {a ^ {4} r ^ {2n + 3}} {2n + 3}}} ∫ r d X X = r − A ln | A + r X | = r − A arsinh A X { displaystyle int { frac {r , dx} {x}} = ra ln vlevo | { frac {a + r} {x}} doprava | = ra , operatorname {arsinh} { frac {a} {x}}} ∫ r 3 d X X = r 3 3 + A 2 r − A 3 ln | A + r X | { displaystyle int { frac {r ^ {3} , dx} {x}} = { frac {r ^ {3}} {3}} + a ^ {2} ra ^ {3} ln left | { frac {a + r} {x}} right |} ∫ r 5 d X X = r 5 5 + A 2 r 3 3 + A 4 r − A 5 ln | A + r X | { displaystyle int { frac {r ^ {5} , dx} {x}} = { frac {r ^ {5}} {5}} + { frac {a ^ {2} r ^ { 3}} {3}} + a ^ {4} ra ^ {5} ln left | { frac {a + r} {x}} right |} ∫ r 7 d X X = r 7 7 + A 2 r 5 5 + A 4 r 3 3 + A 6 r − A 7 ln | A + r X | { displaystyle int { frac {r ^ {7} , dx} {x}} = { frac {r ^ {7}} {7}} + { frac {a ^ {2} r ^ { 5}} {5}} + { frac {a ^ {4} r ^ {3}} {3}} + a ^ {6} ra ^ {7} ln left | { frac {a + r } {x}} doprava |} ∫ d X r = arsinh X A = ln ( X + r A ) { displaystyle int { frac {dx} {r}} = operatorname {arsinh} { frac {x} {a}} = ln left ({ frac {x + r} {a}} že jo)} ∫ d X r 3 = X A 2 r { displaystyle int { frac {dx} {r ^ {3}}} = { frac {x} {a ^ {2} r}}} ∫ X d X r = r { displaystyle int { frac {x , dx} {r}} = r} ∫ X d X r 3 = − 1 r { displaystyle int { frac {x , dx} {r ^ {3}}} = - { frac {1} {r}}} ∫ X 2 d X r = X 2 r − A 2 2 arsinh X A = X 2 r − A 2 2 ln ( X + r A ) { displaystyle int { frac {x ^ {2} , dx} {r}} = { frac {x} {2}} r - { frac {a ^ {2}} {2}} , operatorname {arsinh} { frac {x} {a}} = { frac {x} {2}} r - { frac {a ^ {2}} {2}} ln left ({ frac {x + r} {a}} vpravo)} ∫ d X X r = − 1 A arsinh A X = − 1 A ln | A + r X | { displaystyle int { frac {dx} {xr}} = - { frac {1} {a}} , operatorname {arsinh} { frac {a} {x}} = - { frac { 1} {a}} ln left | { frac {a + r} {x}} right |} Integrály zahrnující s = √X 2 − A 2 Převzít X 2 > A 2 (pro X 2 < A 2 , viz další část):
∫ s d X = 1 2 ( X s − A 2 ln | X + s | ) { displaystyle int s , dx = { frac {1} {2}} vlevo (xs-a ^ {2} ln left | x + s right | right)} ∫ X s d X = 1 3 s 3 { displaystyle int xs , dx = { frac {1} {3}} s ^ {3}} ∫ s d X X = s − | A | arccos | A X | { displaystyle int { frac {s , dx} {x}} = s- | a | arccos left | { frac {a} {x}} vpravo |} ∫ d X s = ln | X + s A | { displaystyle int { frac {dx} {s}} = ln vlevo | { frac {x + s} {a}} vpravo |} Tady ln | X + s A | = sgn ( X ) arcosh | X A | = 1 2 ln ( X + s X − s ) { displaystyle ln left | { frac {x + s} {a}} right | = operatorname {sgn} (x) , operatorname {arcosh} left | { frac {x} {a }} right | = { frac {1} {2}} ln left ({ frac {x + s} {xs}} right)} , kde je kladná hodnota arcosh | X A | { displaystyle operatorname {arcosh} vlevo | { frac {x} {a}} vpravo |} je třeba vzít.
∫ X d X s = s { displaystyle int { frac {x , dx} {s}} = s} ∫ X d X s 3 = − 1 s { displaystyle int { frac {x , dx} {s ^ {3}}} = - { frac {1} {s}}} ∫ X d X s 5 = − 1 3 s 3 { displaystyle int { frac {x , dx} {s ^ {5}}} = - { frac {1} {3s ^ {3}}}} ∫ X d X s 7 = − 1 5 s 5 { displaystyle int { frac {x , dx} {s ^ {7}}} = - { frac {1} {5s ^ {5}}}} ∫ X d X s 2 n + 1 = − 1 ( 2 n − 1 ) s 2 n − 1 { displaystyle int { frac {x , dx} {s ^ {2n + 1}}} = - { frac {1} {(2n-1) s ^ {2n-1}}}} ∫ X 2 m d X s 2 n + 1 = − 1 2 n − 1 X 2 m − 1 s 2 n − 1 + 2 m − 1 2 n − 1 ∫ X 2 m − 2 d X s 2 n − 1 { displaystyle int { frac {x ^ {2m} , dx} {s ^ {2n + 1}}} = - { frac {1} {2n-1}} { frac {x ^ {2m -1}} {s ^ {2n-1}}} + { frac {2m-1} {2n-1}} int { frac {x ^ {2m-2} , dx} {s ^ { 2n-1}}}} ∫ X 2 d X s = X s 2 + A 2 2 ln | X + s A | { displaystyle int { frac {x ^ {2} , dx} {s}} = { frac {xs} {2}} + { frac {a ^ {2}} {2}} ln left | { frac {x + s} {a}} right |} ∫ X 2 d X s 3 = − X s + ln | X + s A | { displaystyle int { frac {x ^ {2} , dx} {s ^ {3}}} = - { frac {x} {s}} + ln left | { frac {x + s} {a}} doprava |} ∫ X 4 d X s = X 3 s 4 + 3 8 A 2 X s + 3 8 A 4 ln | X + s A | { displaystyle int { frac {x ^ {4} , dx} {s}} = { frac {x ^ {3} s} {4}} + { frac {3} {8}} a ^ {2} xs + { frac {3} {8}} a ^ {4} ln left | { frac {x + s} {a}} right |} ∫ X 4 d X s 3 = X s 2 − A 2 X s + 3 2 A 2 ln | X + s A | { displaystyle int { frac {x ^ {4} , dx} {s ^ {3}}} = { frac {xs} {2}} - { frac {a ^ {2} x} { s}} + { frac {3} {2}} a ^ {2} ln left | { frac {x + s} {a}} right |} ∫ X 4 d X s 5 = − X s − 1 3 X 3 s 3 + ln | X + s A | { displaystyle int { frac {x ^ {4} , dx} {s ^ {5}}} = - { frac {x} {s}} - { frac {1} {3}} { frac {x ^ {3}} {s ^ {3}}} + ln left | { frac {x + s} {a}} right |} ∫ X 2 m d X s 2 n + 1 = ( − 1 ) n − m 1 A 2 ( n − m ) ∑ i = 0 n − m − 1 1 2 ( m + i ) + 1 ( n − m − 1 i ) X 2 ( m + i ) + 1 s 2 ( m + i ) + 1 ( n > m ≥ 0 ) { displaystyle int { frac {x ^ {2m} , dx} {s ^ {2n + 1}}} = (- 1) ^ {nm} { frac {1} {a ^ {2 (nm )}}} sum _ {i = 0} ^ {nm-1} { frac {1} {2 (m + i) +1}} {nm-1 vyberte i} { frac {x ^ { 2 (m + i) +1}} {s ^ {2 (m + i) +1}}} qquad { mbox {(}} n> m geq 0 { mbox {)}}} ∫ d X s 3 = − 1 A 2 X s { displaystyle int { frac {dx} {s ^ {3}}} = - { frac {1} {a ^ {2}}} { frac {x} {s}}} ∫ d X s 5 = 1 A 4 [ X s − 1 3 X 3 s 3 ] { displaystyle int { frac {dx} {s ^ {5}}} = { frac {1} {a ^ {4}}} left [{ frac {x} {s}} - { frac {1} {3}} { frac {x ^ {3}} {s ^ {3}}} vpravo]} ∫ d X s 7 = − 1 A 6 [ X s − 2 3 X 3 s 3 + 1 5 X 5 s 5 ] { displaystyle int { frac {dx} {s ^ {7}}} = - { frac {1} {a ^ {6}}} doleva [{ frac {x} {s}} - { frac {2} {3}} { frac {x ^ {3}} {s ^ {3}}} + { frac {1} {5}} { frac {x ^ {5}} {s ^ {5}}} vpravo]} ∫ d X s 9 = 1 A 8 [ X s − 3 3 X 3 s 3 + 3 5 X 5 s 5 − 1 7 X 7 s 7 ] { displaystyle int { frac {dx} {s ^ {9}}} = { frac {1} {a ^ {8}}} left [{ frac {x} {s}} - { frac {3} {3}} { frac {x ^ {3}} {s ^ {3}}} + { frac {3} {5}} { frac {x ^ {5}} {s ^ {5}}} - { frac {1} {7}} { frac {x ^ {7}} {s ^ {7}}} vpravo]} ∫ X 2 d X s 5 = − 1 A 2 X 3 3 s 3 { displaystyle int { frac {x ^ {2} , dx} {s ^ {5}}} = - { frac {1} {a ^ {2}}} { frac {x ^ {3 }} {3s ^ {3}}}} ∫ X 2 d X s 7 = 1 A 4 [ 1 3 X 3 s 3 − 1 5 X 5 s 5 ] { displaystyle int { frac {x ^ {2} , dx} {s ^ {7}}} = { frac {1} {a ^ {4}}} vlevo [{ frac {1} {3}} { frac {x ^ {3}} {s ^ {3}}} - { frac {1} {5}} { frac {x ^ {5}} {s ^ {5}} }že jo]} ∫ X 2 d X s 9 = − 1 A 6 [ 1 3 X 3 s 3 − 2 5 X 5 s 5 + 1 7 X 7 s 7 ] { displaystyle int { frac {x ^ {2} , dx} {s ^ {9}}} = - { frac {1} {a ^ {6}}} vlevo [{ frac {1 } {3}} { frac {x ^ {3}} {s ^ {3}}} - { frac {2} {5}} { frac {x ^ {5}} {s ^ {5} }} + { frac {1} {7}} { frac {x ^ {7}} {s ^ {7}}} vpravo]} Integrály zahrnující u = √A 2 − X 2 ∫ u d X = 1 2 ( X u + A 2 arcsin X A ) ( | X | ≤ | A | ) { displaystyle int u , dx = { frac {1} {2}} vlevo (xu + a ^ {2} arcsin { frac {x} {a}} vpravo) qquad { mbox {(}} | x | leq | a | { mbox {)}}} ∫ X u d X = − 1 3 u 3 ( | X | ≤ | A | ) { displaystyle int xu , dx = - { frac {1} {3}} u ^ {3} qquad { mbox {(}} | x | leq | a | { mbox {)}} } ∫ X 2 u d X = − X 4 u 3 + A 2 8 ( X u + A 2 arcsin X A ) ( | X | ≤ | A | ) { displaystyle int x ^ {2} u , dx = - { frac {x} {4}} u ^ {3} + { frac {a ^ {2}} {8}} (xu + a ^ {2} arcsin { frac {x} {a}}) qquad { mbox {(}} | x | leq | a | { mbox {)}}} ∫ u d X X = u − A ln | A + u X | ( | X | ≤ | A | ) { displaystyle int { frac {u , dx} {x}} = ua ln vlevo | { frac {a + u} {x}} doprava | qquad { mbox {(}} | x | leq | a | { mbox {)}}} ∫ d X u = arcsin X A ( | X | ≤ | A | ) { displaystyle int { frac {dx} {u}} = arcsin { frac {x} {a}} qquad { mbox {(}} | x | leq | a | { mbox {) }}} ∫ X 2 d X u = 1 2 ( − X u + A 2 arcsin X A ) ( | X | ≤ | A | ) { displaystyle int { frac {x ^ {2} , dx} {u}} = { frac {1} {2}} vlevo (-xu + a ^ {2} arcsin { frac { x} {a}} vpravo) qquad { mbox {(}} | x | leq | a | { mbox {)}}} ∫ u d X = 1 2 ( X u − sgn X arcosh | X A | ) (pro | X | ≥ | A | ) { displaystyle int u , dx = { frac {1} {2}} vlevo (xu- operatorname {sgn} x , operatorname {arcosh} left | { frac {x} {a} } right | right) qquad { mbox {(pro}} | x | geq | a | { mbox {)}}} ∫ X u d X = − u ( | X | ≤ | A | ) { displaystyle int { frac {x} {u}} , dx = -u qquad { mbox {(}} | x | leq | a | { mbox {)}}} Integrály zahrnující R = √sekera 2 + bx + C Předpokládejme (sekera 2 + bx + C ) nelze redukovat na následující výraz (px + q )2 pro některé p a q .
∫ d X R = 1 A ln | 2 A R + 2 A X + b | (pro A > 0 ) { displaystyle int { frac {dx} {R}} = { frac {1} { sqrt {a}}} ln left | 2 { sqrt {a}} R + 2ax + b vpravo | qquad { mbox {(pro}} a> 0 { mbox {)}}} ∫ d X R = 1 A arsinh 2 A X + b 4 A C − b 2 (pro 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 {(pro}} a> 0 { mbox {,}} 4ac-b ^ {2}> 0 { mbox {)}}} ∫ d X R = 1 A ln | 2 A X + b | (pro A > 0 , 4 A C − b 2 = 0 ) { displaystyle int { frac {dx} {R}} = { frac {1} { sqrt {a}}} ln | 2ax + b | quad { mbox {(pro}} a> 0 { mbox {,}} 4ac-b ^ {2} = 0 { mbox {)}}} ∫ d X R = − 1 − A arcsin 2 A X + b b 2 − 4 A C (pro 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 {(pro}} a <0 { mbox {,}} 4ac-b ^ {2} <0 { mbox {,}} vlevo | 2ax + b vpravo | <{ 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}} vlevo ({ frac {1 } {R ^ {2}}} + { frac {8a} {4ac-b ^ {2}}} vpravo)} ∫ 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})}}} vlevo ({ 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 | , C > 0 { displaystyle int { frac {dx} {xR}} = - { frac {1} { sqrt {c}}} ln left | { frac {2 { sqrt {c}} R + bx + 2c} {x}} doprava |, ~ c> 0} ∫ d X X R = − 1 C arsinh ( b X + 2 C | X | 4 A C − b 2 ) , C < 0 { displaystyle int { frac {dx} {xR}} = - { frac {1} { sqrt {c}}} operatorname {arsinh} left ({ frac {bx + 2c} {| x | { sqrt {4ac-b ^ {2}}}}} vpravo), ~ c <0} ∫ d X X R = 1 − C arcsin ( b X + 2 C | X | b 2 − 4 A C ) , C < 0 , b 2 − 4 A C > 0 { displaystyle int { frac {dx} {xR}} = { frac {1} { sqrt {-c}}} operatorname {arcsin} left ({ frac {bx + 2c} {| x | { sqrt {b ^ {2} -4ac}}}} right), ~ c <0, b ^ {2} -4ac> 0} ∫ d X X R = − 2 b X ( A X 2 + b X ) , C = 0 { displaystyle int { frac {dx} {xR}} = - { frac {2} {bx}} vlevo ({ sqrt {ax ^ {2} + bx}} vpravo), ~ c = 0} ∫ X 2 R d X = 2 A X − 3 b 4 A 2 R + 3 b 2 − 4 A C 8 A 2 ∫ d X R { displaystyle int { frac {x ^ {2}} {R}} , dx = { frac {2ax-3b} {4a ^ {2}}} R + { frac {3b ^ {2} - 4ac} {8a ^ {2}}} int { frac {dx} {R}}} ∫ d X X 2 R = − R C X − b 2 C ∫ d X X R { displaystyle int { frac {dx} {x ^ {2} R}} = - { frac {R} {cx}} - { frac {b} {2c}} int { frac {dx } {xR}}} ∫ R d X = 2 A X + b 4 A R + 4 A C − b 2 8 A ∫ d X R { displaystyle int R , dx = { frac {2ax + b} {4a}} R + { frac {4ac-b ^ {2}} {8a}} int { frac {dx} {R} }} ∫ X R d X = R 3 3 A − b ( 2 A X + b ) 8 A 2 R − b ( 4 A C − b 2 ) 16 A 2 ∫ d X R { displaystyle int xR , dx = { frac {R ^ {3}} {3a}} - { frac {b (2ax + b)} {8a ^ {2}}} R - { frac { b (4ac-b ^ {2})} {16a ^ {2}}} int { frac {dx} {R}}} ∫ X 2 R d X = 6 A X − 5 b 24 A 2 R 3 + 5 b 2 − 4 A C 16 A 2 ∫ R d X { displaystyle int x ^ {2} R , dx = { frac {6ax-5b} {24a ^ {2}}} R ^ {3} + { frac {5b ^ {2} -4ac} { 16a ^ {2}}} int R , dx} ∫ R X d X = R + b 2 ∫ d X R + C ∫ d X X R { displaystyle int { frac {R} {x}} , dx = R + { frac {b} {2}} int { frac {dx} {R}} + c int { frac { dx} {xR}}} ∫ R X 2 d X = − R X + A ∫ d X R + b 2 ∫ d X X R { displaystyle int { frac {R} {x ^ {2}}} , dx = - { frac {R} {x}} + a int { frac {dx} {R}} + { frac {b} {2}} int { frac {dx} {xR}}} ∫ X 2 d X R 3 = ( 2 b 2 − 4 A C ) X + 2 b C A ( 4 A C − b 2 ) R + 1 A ∫ d X R { displaystyle int { frac {x ^ {2} , dx} {R ^ {3}}} = { frac {(2b ^ {2} -4ac) x + 2bc} {a (4ac-b ^ {2}) R}} + { frac {1} {a}} int { frac {dx} {R}}} Integrály zahrnující S = √sekera + b ∫ S d X = 2 S 3 3 A { displaystyle int S , dx = { frac {2S ^ {3}} {3a}}} ∫ d X S = 2 S A { displaystyle int { frac {dx} {S}} = { frac {2S} {a}}} ∫ d X X S = { − 2 b arcoth ( S b ) (pro b > 0 , A X > 0 ) − 2 b artanh ( S b ) (pro b > 0 , A X < 0 ) 2 − b arktan ( S − b ) (pro b < 0 ) { displaystyle int { frac {dx} {xS}} = { begin {případy} - { dfrac {2} { sqrt {b}}} operatorname {arcoth} left ({ dfrac {S } { sqrt {b}}} right) & { mbox {(for}} b> 0, quad ax> 0 { mbox {)}} - { dfrac {2} { sqrt { b}}} operatorname {artanh} left ({ dfrac {S} { sqrt {b}}} right) & { mbox {(for}} b> 0, quad ax <0 { mbox {)}} { dfrac {2} { sqrt {-b}}} arctan left ({ dfrac {S} { sqrt {-b}}} right) & { mbox {( for}} b <0 { mbox {)}} konec {případů}}} ∫ S X d X = { 2 ( S − b arcoth ( S b ) ) (pro b > 0 , A X > 0 ) 2 ( S − b artanh ( S b ) ) (pro b > 0 , A X < 0 ) 2 ( S − − b arktan ( S − b ) ) (pro b < 0 ) { displaystyle int { frac {S} {x}} , dx = { začátek {případů} 2 left (S - { sqrt {b}} , operatorname {arcoth} left ({ dfrac {S} { sqrt {b}}} right) right) & { mbox {(for}} b> 0, quad ax> 0 { mbox {)}} 2 left (S - { sqrt {b}} , operatorname {artanh} left ({ dfrac {S} { sqrt {b}}} right) right) & { mbox {(for}} b> 0 , quad ax <0 { mbox {)}} 2 left (S - { sqrt {-b}} arctan left ({ dfrac {S} { sqrt {-b}}} ) right) right) & { mbox {(for}} b <0 { mbox {)}} end {cases}}} ∫ X n S d X = 2 A ( 2 n + 1 ) ( X n S − b n ∫ X n − 1 S d X ) { displaystyle int { frac {x ^ {n}} {S}} , dx = { frac {2} {a (2n + 1)}} vlevo (x ^ {n} S-bn int { frac {x ^ {n-1}} {S}} , dx right)} ∫ X n S d X = 2 A ( 2 n + 3 ) ( X n S 3 − n b ∫ X n − 1 S d X ) { displaystyle int x ^ {n} S , dx = { frac {2} {a (2n + 3)}} vlevo (x ^ {n} S ^ {3} -nb int x ^ { n-1} S , dx vpravo)} ∫ 1 X n S d X = − 1 b ( n − 1 ) ( S X n − 1 + ( n − 3 2 ) A ∫ d X X n − 1 S ) { displaystyle int { frac {1} {x ^ {n} S}} , dx = - { frac {1} {b (n-1)}} left ({ frac {S} { x ^ {n-1}}} + left (n - { frac {3} {2}} right) a int { frac {dx} {x ^ {n-1} S}} right )} Reference Peirce, Benjamin Osgood (1929) [1899]. „Kapitola 3“. Krátká tabulka integrálů (3. přepracované vydání). Boston: Ginn and Co., str. 16–30. Milton Abramowitz a Irene A. Stegun, eds., Příručka matematických funkcí se vzorci, grafy a matematickými tabulkami 1972, Dover: New York. (Vidět Kapitola 3 .) Gradshteyn, Izrail Solomonovich ; Ryzhik, Iosif Moiseevich ; Geronimus, Jurij Veniaminovič ; Tseytlin, Michail Yulyevich ; Jeffrey, Alan (2015) [říjen 2014]. Zwillinger, Daniel; Moll, Victor Hugo (eds.). Tabulka integrálů, sérií a produktů . Přeložil Scripta Technica, Inc. (8. vydání). Academic Press, Inc. ISBN 978-0-12-384933-5 . LCCN 2014010276 . (Několik předchozích vydání také.)