Lista de integrais de funcións irracionais

A seguinte é unha lista de integrais de funcións irracionais. Integrais con r √x2+a2
∫ r d x
12 ( 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
14 x r 3 + 183 a 2 x r + 38 a 4 ln ⁡ ( x + r ) {\displaystyle \int r^{3}\; dx
{\frac {1}{4}}xr^{3}+{\frac {1}{8}}3a^{2}xr+{\frac {3}{8}}a^{4}\ln \left(x+r\right)} ∫ r 5 d x
16 x r 5 + 524 a 2 x r 3 + 516 a 4 x r + 516 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 33 {\displaystyle \int xr\; dx
{\frac {r^{3}}{3}}} ∫ x r 3 d x
r 55 {\displaystyle \int xr^{3}\; dx
{\frac {r^{5}}{5}}} ∫ x r 2 n + 1 d x
r 2 n + 32 n + 3 {\displaystyle \int xr^{2n+1}\; dx
{\frac {r^{2n+3}}{2n+3}}} ∫ x 2 r d x
x r 34 − a 2 x r 8 − a 48 ln ⁡ ( x + r ) {\displaystyle \int x^{2}r\; dx
{\frac {xr^{3}}{4}}-{\frac {a^{2}xr}{8}}-{\frac {a^{4}}{8}}\ln \left(x+r\right)} ∫ x 2 r 3 d x
x r 56 − a 2 x r 324 − a 4 x r 16 − a 616 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 55 − a 2 r 33 {\displaystyle \int x^{3}r\; dx
{\frac {r^{5}}{5}}-{\frac {a^{2}r^{3}}{3}}} ∫ x 3 r 3 d x
r 77 − a 2 r 55 {\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 + 52 n + 5 − a 3 r 2 n + 32 n + 3 {\displaystyle \int x^{3}r^{2n+1}\; dx
{\frac {r^{2n+5}}{2n+5}}-{\frac {a^{3}r^{2n+3}}{2n+3}}} ∫ x 4 r d x
x 3 r 36 − a 2 x r 38 + a 4 x r 16 + a 616 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 58 − a 2 x r 516 + a 4 x r 364 + 3 a 6 x r 128 + 3 a 8128 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 \left(x+r\right)} ∫ x 5 r d x
r 77 − 2 a 2 r 55 + a 4 r 33 {\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 99 − 2 a 2 r 77 + a 4 r 55 {\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 + 72 n + 7 − 2 a 2 r 2 n + 52 n + 5 + a 4 r 2 n + 32 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 sinh − 1 ⁡ a x {\displaystyle \int {\frac {r\; dx}{x}}
r-a\ln \left|{\frac {a+r}{x}}\right|
r-a\sinh ^{-1}{\frac {a}{x}}} ∫ r 3 d x x
r 33 + a 2 r − a 3 ln ⁡ | a + r x | {\displaystyle \int {\frac {r^{3}\; dx}{x}}
{\frac {r^{3}}{3}}+a^{2}r-a^{3}\ln \left|{\frac {a+r}{x}}\right|} ∫ r 5 d x x
r 55 + a 2 r 33 + 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}r-a^{5}\ln \left|{\frac {a+r}{x}}\right|} ∫ r 7 d x x
r 77 + a 2 r 55 + a 4 r 33 + 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}r-a^{7}\ln \left|{\frac {a+r}{x}}\right|} ∫ d x r
sinh − 1 ⁡ x a
ln ⁡ | x + r | {\displaystyle \int {\frac {dx}{r}}
\sinh ^{-1}{\frac {x}{a}}
\ln \left|x+r\right|} ∫ 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 22 sinh − 1 ⁡ x a
x 2 r − a 22 ln ⁡ | x + r | {\displaystyle \int {\frac {x^{2}\; dx}{r}}
{\frac {x}{2}}r-{\frac {a^{2}}{2}}\, \sinh ^{-1}{\frac {x}{a}}
{\frac {x}{2}}r-{\frac {a^{2}}{2}}\ln \left|x+r\right|} ∫ d x x r
− 1 a sinh − 1 ⁡ a x
− 1 a ln ⁡ | a + r x | {\displaystyle \int {\frac {dx}{xr}}
-{\frac {1}{a}}\, \sinh ^{-1}{\frac {a}{x}}
-{\frac {1}{a}}\ln \left|{\frac {a+r}{x}}\right|} Integrais con s √x2-a2
∫ x s d x
13 s 3 {\displaystyle \int xs\; dx
{\frac {1}{3}}s^{3}} ∫ s d x x
s − a cos − 1 ⁡ | a x | {\displaystyle \int {\frac {s\; dx}{x}}
s-a\cos ^{-1}\left|{\frac {a}{x}}\right|} ∫ d x s
∫ d x x 2 − a 2
ln ⁡ | x + s a | {\displaystyle \int {\frac {dx}{s}}
\int {\frac {dx}{\sqrt {x^{2}-a^{2}}}}
\ln \left|{\frac {x+s}{a}}\right|} ∫ 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
− 13 s 3 {\displaystyle \int {\frac {x\; dx}{s^{5}}}
-{\frac {1}{3s^{3}}}} ∫ x d x s 7
− 15 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
− 12 n − 1 x 2 m − 1 s 2 n − 1 + 2 m − 12 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 22 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}}\right|} ∫ x 4 d x s
x 3 s 4 + 38 a 2 x s + 38 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 + 32 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 − 13 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 − 112 ( 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)^{n-m}{\frac {1}{a^{2(n-m)}}}\sum _{i
0}^{n-m-1}{\frac {1}{2(m+i)+1}}{n-m-1 \choose 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 − 13 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}}}\right]} ∫ d x s 7
− 1 a 6 [ x s − 23 x 3 s 3 + 15 x 5 s 5 ] {\displaystyle \int {\frac {dx}{s^{7}}}
-{\frac {1}{a^{6}}}\left[{\frac {x}{s}}-{\frac {2}{3}}{\frac {x^{3}}{s^{3}}}+{\frac {1}{5}}{\frac {x^{5}}{s^{5}}}\right]} ∫ d x s 9
1 a 8 [ x s − 33 x 3 s 3 + 35 x 5 s 5 − 17 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}}}\right]} ∫ x 2 d x s 5
− 1 a 2 x 33 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 [ 13 x 3 s 3 − 15 x 5 s 5 ] {\displaystyle \int {\frac {x^{2}\; dx}{s^{7}}}
{\frac {1}{a^{4}}}\left[{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}-{\frac {1}{5}}{\frac {x^{5}}{s^{5}}}\right]} ∫ x 2 d x s 9
− 1 a 6 [ 13 x 3 s 3 − 25 x 5 s 5 + 17 x 7 s 7 ] {\displaystyle \int {\frac {x^{2}\; dx}{s^{9}}}
-{\frac {1}{a^{6}}}\left[{\frac {1}{3}}{\frac {x^{3}}{s^{3}}}-{\frac {2}{5}}{\frac {x^{5}}{s^{5}}}+{\frac {1}{7}}{\frac {x^{7}}{s^{7}}}\right]} Integrais con t √a2-x2
∫ t d x
12 ( x t + a 2 sin − 1 ⁡ x a ) ( | x | ≤ | a | ) {\displaystyle \int t\; dx
{\frac {1}{2}}\left(xt+a^{2}\sin ^{-1}{\frac {x}{a}}\right)\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}} ∫ x t d x
− 13 t 3 ( | x | ≤ | a | ) {\displaystyle \int xt\; dx
-{\frac {1}{3}}t^{3}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}} ∫ t d x x
t − a ln ⁡ | a + t x | ( | x | ≤ | a | ) {\displaystyle \int {\frac {t\; dx}{x}}
t-a\ln \left|{\frac {a+t}{x}}\right|\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}} ∫ d x t
sin − 1 ⁡ x a ( | x | ≤ | a | ) {\displaystyle \int {\frac {dx}{t}}
\sin ^{-1}{\frac {x}{a}}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}} ∫ x 2 d x t
− x 2 t + a 22 sin − 1 ⁡ x a ( | x | ≤ | a | ) {\displaystyle \int {\frac {x^{2}\; dx}{t}}
-{\frac {x}{2}}t+{\frac {a^{2}}{2}}\sin ^{-1}{\frac {x}{a}}\qquad {\mbox{(}}|x|\leq |a|{\mbox{)}}} ∫ t d x
12 ( x t − sgn ⁡ x cosh − 1 ⁡ | x a | ) ( | x | ≥ | a | ) {\displaystyle \int t\; dx
{\frac {1}{2}}\left(xt-\operatorname {sgn} x\, \cosh ^{-1}\left|{\frac {x}{a}}\right|\right)\qquad {\mbox{(}}|x|\geq |a|{\mbox{)}}} Integrais con R √ax2+bx+c
∫ d x a x 2 + b x + c
1 a ln ⁡ | 2 a R + 2 a x + b | (para a > 0 ) {\displaystyle \int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}
{\frac {1}{\sqrt {a}}}\ln \left|2{\sqrt {a}}R+2ax+b\right|\qquad {\mbox{(para }}a>0{\mbox{)}}} ∫ d x a x 2 + b x + c
1 a sinh − 1 ⁡ 2 a x + b 4 a c − b 2 (para a > 0, 4 a c − b 2 > 0 ) {\displaystyle \int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}
{\frac {1}{\sqrt {a}}}\, \sinh ^{-1}{\frac {2ax+b}{\sqrt {4ac-b^{2}}}}\qquad {\mbox{(para }}a>0{\mbox{, }}4ac-b^{2}>0{\mbox{)}}} ∫ d x a x 2 + b x + c
1 a ln ⁡ | 2 a x + b | (para a > 0, 4 a c − b 2
0 ) {\displaystyle \int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}
{\frac {1}{\sqrt {a}}}\ln |2ax+b|\quad {\mbox{(para }}a>0{\mbox{, }}4ac-b^{2}
0{\mbox{)}}} ∫ d x a x 2 + b x + c
− 1 − a arcsin ⁡ 2 a x + b b 2 − 4 a c (para a < 0, 4 a c − b 2 < 0 ) {\displaystyle \int {\frac {dx}{\sqrt {ax^{2}+bx+c}}}
-{\frac {1}{\sqrt {-a}}}\arcsin {\frac {2ax+b}{\sqrt {b^{2}-4ac}}}\qquad {\mbox{(para }}a<0{\mbox{, }}4ac-b^{2}<0{\mbox{)}}} ∫ d x ( a x 2 + b x + c ) 3
4 a x + 2 b ( 4 a c − b 2 ) R {\displaystyle \int {\frac {dx}{\sqrt {(ax^{2}+bx+c)^{3}}}}
{\frac {4ax+2b}{(4ac-b^{2}){\sqrt {R}}}}} ∫ d x ( a x 2 + b x + c ) 5
4 a x + 2 b 3 ( 4 a c − b 2 ) R ( 1 R + 8 a 4 a c − b 2 ) {\displaystyle \int {\frac {dx}{\sqrt {(ax^{2}+bx+c)^{5}}}}
{\frac {4ax+2b}{3(4ac-b^{2}){\sqrt {R}}}}\left({\frac {1}{R}}+{\frac {8a}{4ac-b^{2}}}\right)} ∫ d x ( a x 2 + b x + c ) 2 n + 1
4 a x + 2 b ( 2 n − 1 ) ( 4 a c − b 2 ) R ( 2 n − 1 ) / 2 + 8 a ( n − 1 ) ( 2 n − 1 ) ( 4 a c − b 2 ) ∫ d x R ( 2 n − 1 ) / 2 {\displaystyle \int {\frac {dx}{\sqrt {(ax^{2}+bx+c)^{2n+1}}}}
{\frac {4ax+2b}{(2n-1)(4ac-b^{2})R^{(2n-1)/2}}}+{\frac {8a(n-1)}{(2n-1)(4ac-b^{2})}}\int {\frac {dx}{R^{(2n-1)/2}}}} ∫ x d x a x 2 + b x + c
R a − b 2 a ∫ d x R {\displaystyle \int {\frac {x\; dx}{\sqrt {ax^{2}+bx+c}}}
{\frac {\sqrt {R}}{a}}-{\frac {b}{2a}}\int {\frac {dx}{\sqrt {R}}}} ∫ x d x ( a x 2 + b x + c ) 3
− 2 b x + 4 c ( 4 a c − b 2 ) R {\displaystyle \int {\frac {x\; dx}{\sqrt {(ax^{2}+bx+c)^{3}}}}
-{\frac {2bx+4c}{(4ac-b^{2}){\sqrt {R}}}}} ∫ x d x ( a x 2 + b x + c ) 2 n + 1
− 1 ( 2 n − 1 ) a R ( 2 n − 1 ) / 2 − b 2 a ∫ d x R ( 2 n + 1 ) / 2 {\displaystyle \int {\frac {x\; dx}{\sqrt {(ax^{2}+bx+c)^{2n+1}}}}
-{\frac {1}{(2n-1)aR^{(2n-1)/2}}}-{\frac {b}{2a}}\int {\frac {dx}{R^{(2n+1)/2}}}} ∫ d x x a x 2 + b x + c
− 1 c ln ⁡ ( 2 c R + b x + 2 c x ) (para c > 0 ) {\displaystyle \int {\frac {dx}{x{\sqrt {ax^{2}+bx+c}}}}
-{\frac {1}{\sqrt {c}}}\ln \left({\frac {2{\sqrt {cR}}+bx+2c}{x}}\right)\qquad {\mbox{(para }}c>0{\mbox{)}}} ∫ d x x a x 2 + b x + c
1 − c sin − 1 ⁡ ( b x + 2 c | x | b 2 − 4 a c ) (para c < 0 ) {\displaystyle \int {\frac {dx}{x{\sqrt {ax^{2}+bx+c}}}}
{\frac {1}{\sqrt {-c}}}\sin ^{-1}\left({\frac {bx+2c}{|x|{\sqrt {b^{2}-4ac}}}}\right)\qquad {\mbox{(para }}c<0{\mbox{)}}} ∫ d x x a x 2 + b x + c
− 1 c sinh − 1 ⁡ ( b x + 2 c | x | 4 a c − b 2 ) {\displaystyle \int {\frac {dx}{x{\sqrt {ax^{2}+bx+c}}}}
-{\frac {1}{\sqrt {c}}}\sinh ^{-1}\left({\frac {bx+2c}{|x|{\sqrt {4ac-b^{2}}}}}\right)} Integrais con √ax+b
∫ d x x a x + b
− 2 b tanh − 1 ⁡ a x + b b {\displaystyle \int {\frac {dx}{x{\sqrt {ax+b}}}}\,
\, {\frac {-2}{\sqrt {b}}}\tanh ^{-1}{\sqrt {\frac {ax+b}{b}}}} ∫ a x + b x d x
2 ( a x + b − b tanh − 1 ⁡ a x + b b ) {\displaystyle \int {\frac {\sqrt {ax+b}}{x}}\, dx\;
\; 2\left({\sqrt {ax+b}}-{\sqrt {b}}\tanh ^{-1}{\sqrt {\frac {ax+b}{b}}}\right)} ∫ x n a x + b d x
2 a ( 2 n + 1 ) ( x n a x + b − b n ∫ x n − 1 a x + b d x ) {\displaystyle \int {\frac {x^{n}}{\sqrt {ax+b}}}\, dx\;
\; {\frac {2}{a(2n+1)}}\left(x^{n}{\sqrt {ax+b}}-bn\int {\frac {x^{n-1}}{\sqrt {ax+b}}}\, dx\right)} ∫ x n a x + b d x
22 n + 1 ( x n + 1 a x + b + b x n a x + b − n b ∫ x n − 1 a x + b d x ) {\displaystyle \int x^{n}{\sqrt {ax+b}}\, dx\;
\; {\frac {2}{2n+1}}\left(x^{n+1}{\sqrt {ax+b}}+bx^{n}{\sqrt {ax+b}}-nb\int x^{n-1}{\sqrt {ax+b}}\, dx\right)}