[ تعرٌف على ] قائمة تكاملات الدوال الزائدية
تم النشر اليوم [dadate] | قائمة تكاملات الدوال الزائدية
التكاملات الأخرى
تكاملات دوال الظل، وظل التمام، والقاطع، وقاطع التمام الزائدية ∫
tanh
x d
x
=
ln
cosh
x
+
C
{\displaystyle \int \tanh x\,dx=\ln \cosh x+C}
∫ tanh 2
a
x d
x
=
x
− tanh
a
x a
+
C
{\displaystyle \int \tanh ^{2}ax\,dx=x-{\frac {\tanh ax}{a}}+C}
∫ tanh n
a
x d
x
=
−
1 a
(
n
−
1
)
tanh n
−
1
a
x
+
∫ tanh n
−
2
a
x d
x (for
n
≠
1
)
{\displaystyle \int \tanh ^{n}ax\,dx=-{\frac {1}{a(n-1)}}\tanh ^{n-1}ax+\int \tanh ^{n-2}ax\,dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}
∫
coth
x d
x
=
ln
| sinh
x | +
C
, for x
≠
0
{\displaystyle \int \coth x\,dx=\ln |\sinh x|+C,{\text{ for }}x\neq 0}
∫ coth n
a
x d
x
=
−
1 a
(
n
−
1
)
coth n
−
1
a
x
+
∫ coth n
−
2
a
x d
x (for
n
≠
1
)
{\displaystyle \int \coth ^{n}ax\,dx=-{\frac {1}{a(n-1)}}\coth ^{n-1}ax+\int \coth ^{n-2}ax\,dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}
∫
sech x d
x
=
arctan (
sinh
x
)
+
C
{\displaystyle \int \operatorname {sech} \,x\,dx=\arctan \,(\sinh x)+C}
∫
csch x d
x
=
ln
| tanh
x
2 | +
C
, for x
≠
0
{\displaystyle \int \operatorname {csch} \,x\,dx=\ln \left|\tanh {x \over 2}\right|+C,{\text{ for }}x\neq 0} التكاملات التي تحتوي على دالتي الجيب وجيب التمام الزائدية ∫
(
cosh
a
x
)
(
sinh
b
x
) d
x
=
1
a 2
− b 2 (
a
(
sinh
a
x
)
(
sinh
b
x
)
−
b
(
cosh
a
x
)
(
cosh
b
x
)
)
+
C (for a 2
≠ b 2
)
{\displaystyle \int (\cosh ax)(\sinh bx)\,dx={\frac {1}{a^{2}-b^{2}}}{\big (}a(\sinh ax)(\sinh bx)-b(\cosh ax)(\cosh bx){\big )}+C\qquad {\mbox{(for }}a^{2}\neq b^{2}{\mbox{)}}}
∫
cosh n
a
x sinh m
a
x d
x
=
cosh n
−
1
a
x
a
(
n
−
m
) sinh m
−
1
a
x + n
−
1
n
−
m ∫
cosh n
−
2
a
x sinh m
a
x d
x (for
m
≠
n
)
{\displaystyle \int {\frac {\cosh ^{n}ax}{\sinh ^{m}ax}}dx={\frac {\cosh ^{n-1}ax}{a(n-m)\sinh ^{m-1}ax}}+{\frac {n-1}{n-m}}\int {\frac {\cosh ^{n-2}ax}{\sinh ^{m}ax}}dx\qquad {\mbox{(for }}m\neq n{\mbox{)}}} أيضًا:
∫
cosh n
a
x sinh m
a
x d
x
=
−
cosh n
+
1
a
x
a
(
m
−
1
) sinh m
−
1
a
x + n
−
m
+
2
m
−
1 ∫
cosh n
a
x sinh m
−
2
a
x d
x (for
m
≠
1
)
{\displaystyle \int {\frac {\cosh ^{n}ax}{\sinh ^{m}ax}}dx=-{\frac {\cosh ^{n+1}ax}{a(m-1)\sinh ^{m-1}ax}}+{\frac {n-m+2}{m-1}}\int {\frac {\cosh ^{n}ax}{\sinh ^{m-2}ax}}dx\qquad {\mbox{(for }}m\neq 1{\mbox{)}}}
∫
cosh n
a
x sinh m
a
x d
x
=
−
cosh n
−
1
a
x
a
(
m
−
1
) sinh m
−
1
a
x + n
−
1
m
−
1 ∫
cosh n
−
2
a
x sinh m
−
2
a
x d
x (for
m
≠
1
)
{\displaystyle \int {\frac {\cosh ^{n}ax}{\sinh ^{m}ax}}dx=-{\frac {\cosh ^{n-1}ax}{a(m-1)\sinh ^{m-1}ax}}+{\frac {n-1}{m-1}}\int {\frac {\cosh ^{n-2}ax}{\sinh ^{m-2}ax}}dx\qquad {\mbox{(for }}m\neq 1{\mbox{)}}}
∫
sinh m
a
x cosh n
a
x d
x
=
sinh m
−
1
a
x
a
(
m
−
n
) cosh n
−
1
a
x + m
−
1
n
−
m ∫
sinh m
−
2
a
x cosh n
a
x d
x (for
m
≠
n
)
{\displaystyle \int {\frac {\sinh ^{m}ax}{\cosh ^{n}ax}}dx={\frac {\sinh ^{m-1}ax}{a(m-n)\cosh ^{n-1}ax}}+{\frac {m-1}{n-m}}\int {\frac {\sinh ^{m-2}ax}{\cosh ^{n}ax}}dx\qquad {\mbox{(for }}m\neq n{\mbox{)}}}
∫
sinh m
a
x cosh n
a
x d
x
=
sinh m
+
1
a
x
a
(
n
−
1
) cosh n
−
1
a
x + m
−
n
+
2
n
−
1 ∫
sinh m
a
x cosh n
−
2
a
x d
x (for
n
≠
1
)
{\displaystyle \int {\frac {\sinh ^{m}ax}{\cosh ^{n}ax}}dx={\frac {\sinh ^{m+1}ax}{a(n-1)\cosh ^{n-1}ax}}+{\frac {m-n+2}{n-1}}\int {\frac {\sinh ^{m}ax}{\cosh ^{n-2}ax}}dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}}
∫
sinh m
a
x cosh n
a
x d
x
=
−
sinh m
−
1
a
x
a
(
n
−
1
) cosh n
−
1
a
x + m
−
1
n
−
1 ∫
sinh m
−
2
a
x cosh n
−
2
a
x d
x (for
n
≠
1
)
{\displaystyle \int {\frac {\sinh ^{m}ax}{\cosh ^{n}ax}}dx=-{\frac {\sinh ^{m-1}ax}{a(n-1)\cosh ^{n-1}ax}}+{\frac {m-1}{n-1}}\int {\frac {\sinh ^{m-2}ax}{\cosh ^{n-2}ax}}dx\qquad {\mbox{(for }}n\neq 1{\mbox{)}}} التكاملات التي تحتوي على الدوال الزائدية والمثلثية ∫
sinh
(
a
x
+
b
)
sin
(
c
x
+
d
) d
x
=
a
a 2
+ c 2 cosh
(
a
x
+
b
)
sin
(
c
x
+
d
)
−
c
a 2
+ c 2 sinh
(
a
x
+
b
)
cos
(
c
x
+
d
)
+
C
{\displaystyle \int \sinh(ax+b)\sin(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\cosh(ax+b)\sin(cx+d)-{\frac {c}{a^{2}+c^{2}}}\sinh(ax+b)\cos(cx+d)+C}
∫
sinh
(
a
x
+
b
)
cos
(
c
x
+
d
) d
x
=
a
a 2
+ c 2 cosh
(
a
x
+
b
)
cos
(
c
x
+
d
)
+
c
a 2
+ c 2 sinh
(
a
x
+
b
)
sin
(
c
x
+
d
)
+
C
{\displaystyle \int \sinh(ax+b)\cos(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\cosh(ax+b)\cos(cx+d)+{\frac {c}{a^{2}+c^{2}}}\sinh(ax+b)\sin(cx+d)+C}
∫
cosh
(
a
x
+
b
)
sin
(
c
x
+
d
) d
x
=
a
a 2
+ c 2 sinh
(
a
x
+
b
)
sin
(
c
x
+
d
)
−
c
a 2
+ c 2 cosh
(
a
x
+
b
)
cos
(
c
x
+
d
)
+
C
{\displaystyle \int \cosh(ax+b)\sin(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\sinh(ax+b)\sin(cx+d)-{\frac {c}{a^{2}+c^{2}}}\cosh(ax+b)\cos(cx+d)+C}
∫
cosh
(
a
x
+
b
)
cos
(
c
x
+
d
) d
x
=
a
a 2
+ c 2 sinh
(
a
x
+
b
)
cos
(
c
x
+
d
)
+
c
a 2
+ c 2 cosh
(
a
x
+
b
)
sin
(
c
x
+
d
)
+
C
{\displaystyle \int \cosh(ax+b)\cos(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\sinh(ax+b)\cos(cx+d)+{\frac {c}{a^{2}+c^{2}}}\cosh(ax+b)\sin(cx+d)+C}
التكاملات التي تحتوي على دالة الجيب الزائدية
∫
sinh
a
x d
x
=
1
a
cosh
a
x
+
C {\displaystyle \int \sinh ax\,dx={\frac {1}{a}}\cosh ax+C\,}
∫ sinh 2
a
x d
x
=
1 4
a sinh
2
a
x
−
x
2
+
C {\displaystyle \int \sinh ^{2}ax\,dx={\frac {1}{4a}}\sinh 2ax-{\frac {x}{2}}+C\,}
∫ sinh n
a
x d
x
=
1 a
n
sinh n
−
1
a
x
cosh
a
x
− n
−
1 n
∫ sinh n
−
2
a
x d
x (
n
>
0
) {\displaystyle \int \sinh ^{n}ax\,dx={\frac {1}{an}}\sinh ^{n-1}ax\cosh ax-{\frac {n-1}{n}}\int \sinh ^{n-2}ax\,dx\qquad {\mbox{(}}n>0{\mbox{)}}\,}
∫ sinh n
a
x d
x
=
1 a
(
n
+
1
)
sinh n
+
1
a
x
cosh
a
x
− n
+
2
n
+
1 ∫ sinh n
+
2
a
x d
x (
n
<
0
,
n
≠
−
1
) {\displaystyle \int \sinh ^{n}ax\,dx={\frac {1}{a(n+1)}}\sinh ^{n+1}ax\cosh ax-{\frac {n+2}{n+1}}\int \sinh ^{n+2}ax\,dx\qquad {\mbox{(}}n<0{\mbox{, }}n\neq -1{\mbox{)}}\,}
∫ d
x
sinh
a
x =
1
a
ln
| tanh
a
x 2 | +
C {\displaystyle \int {\frac {dx}{\sinh ax}}={\frac {1}{a}}\ln \left|\tanh {\frac {ax}{2}}\right|+C\,}
∫ d
x
sinh
a
x =
1
a
ln
| cosh
a
x
−
1
sinh
a
x | +
C {\displaystyle \int {\frac {dx}{\sinh ax}}={\frac {1}{a}}\ln \left|{\frac {\cosh ax-1}{\sinh ax}}\right|+C\,}
∫ d
x
sinh
a
x =
1
a
ln
| sinh
a
x
cosh
a
x
+
1 | +
C {\displaystyle \int {\frac {dx}{\sinh ax}}={\frac {1}{a}}\ln \left|{\frac {\sinh ax}{\cosh ax+1}}\right|+C\,}
∫ d
x
sinh
a
x =
1
a
ln
| cosh
a
x
−
1
cosh
a
x
+
1 | +
C {\displaystyle \int {\frac {dx}{\sinh ax}}={\frac {1}{a}}\ln \left|{\frac {\cosh ax-1}{\cosh ax+1}}\right|+C\,}
∫ d
x sinh n
a
x =
− cosh
a
x
a
(
n
−
1
) sinh n
−
1
a
x − n
−
2
n
−
1 ∫ d
x sinh n
−
2
a
x
(
n
≠
1
) {\displaystyle \int {\frac {dx}{\sinh ^{n}ax}}=-{\frac {\cosh ax}{a(n-1)\sinh ^{n-1}ax}}-{\frac {n-2}{n-1}}\int {\frac {dx}{\sinh ^{n-2}ax}}\qquad {\mbox{(}}n\neq 1{\mbox{)}}\,}
∫
x
sinh
a
x d
x
=
1
a
x
cosh
a
x
−
1 a 2
sinh
a
x
+
C {\displaystyle \int x\sinh ax\,dx={\frac {1}{a}}x\cosh ax-{\frac {1}{a^{2}}}\sinh ax+C\,}
∫
sinh
(
a
x
+
b
)
sin
(
c
x
+
d
) d
x
=
a
a 2
+ c 2 cosh
(
a
x
+
b
)
sin
(
c
x
+
d
)
−
c
a 2
+ c 2 sinh
(
a
x
+
b
)
cos
(
c
x
+
d
)
+
C {\displaystyle \int \sinh(ax+b)\sin(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\cosh(ax+b)\sin(cx+d)-{\frac {c}{a^{2}+c^{2}}}\sinh(ax+b)\cos(cx+d)+C\,}
∫
sinh
(
a
x
+
b
)
cos
(
c
x
+
d
) d
x
=
a
a 2
+ c 2 cosh
(
a
x
+
b
)
cos
(
c
x
+
d
)
+
c
a 2
+ c 2 sinh
(
a
x
+
b
)
sin
(
c
x
+
d
)
+
C {\displaystyle \int \sinh(ax+b)\cos(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\cosh(ax+b)\cos(cx+d)+{\frac {c}{a^{2}+c^{2}}}\sinh(ax+b)\sin(cx+d)+C\,}
∫
sinh
a
x
sinh
b
x d
x
=
1
a 2
− b 2 (
a
sinh
b
x
cosh
a
x
−
b
cosh
b
x
sinh
a
x
)
+
C ( a 2
≠ b 2
) {\displaystyle \int \sinh ax\sinh bx\,dx={\frac {1}{a^{2}-b^{2}}}(a\sinh bx\cosh ax-b\cosh bx\sinh ax)+C\qquad {\mbox{(}}a^{2}\neq b^{2}{\mbox{)}}\,}
التكاملات التي تحتوي على دالة جيب التمام الزائدية
∫
cosh
a
x d
x
=
1
a
sinh
a
x
+
C {\displaystyle \int \cosh ax\,dx={\frac {1}{a}}\sinh ax+C\,}
∫ cosh 2
a
x d
x
=
1 4
a sinh
2
a
x
+
x
2
+
C {\displaystyle \int \cosh ^{2}ax\,dx={\frac {1}{4a}}\sinh 2ax+{\frac {x}{2}}+C\,}
∫ d
x
cosh
a
x =
2
a
arctan
e a
x
+
C {\displaystyle \int {\frac {dx}{\cosh ax}}={\frac {2}{a}}\arctan e^{ax}+C\,}
∫ cosh n
a
x d
x
=
1 a
n sinh
a
x cosh n
−
1
a
x
+ n
−
1 n
∫ cosh n
−
2
a
x d
x (
n
>
0
) {\displaystyle \int \cosh ^{n}ax\,dx={\frac {1}{an}}\sinh ax\cosh ^{n-1}ax+{\frac {n-1}{n}}\int \cosh ^{n-2}ax\,dx\qquad {\mbox{(}}n>0{\mbox{)}}\,}
∫ cosh n
a
x d
x
=
−
1 a
(
n
+
1
) sinh
a
x cosh n
+
1
a
x
− n
+
2
n
+
1 ∫ cosh n
+
2
a
x d
x (
n
<
0
,
n
≠
−
1
) {\displaystyle \int \cosh ^{n}ax\,dx=-{\frac {1}{a(n+1)}}\sinh ax\cosh ^{n+1}ax-{\frac {n+2}{n+1}}\int \cosh ^{n+2}ax\,dx\qquad {\mbox{(}}n<0{\mbox{, }}n\neq -1{\mbox{)}}\,}
∫ d
x cosh n
a
x = sinh
a
x
a
(
n
−
1
) cosh n
−
1
a
x + n
−
2
n
−
1 ∫ d
x cosh n
−
2
a
x
(
n
≠
1
) {\displaystyle \int {\frac {dx}{\cosh ^{n}ax}}={\frac {\sinh ax}{a(n-1)\cosh ^{n-1}ax}}+{\frac {n-2}{n-1}}\int {\frac {dx}{\cosh ^{n-2}ax}}\qquad {\mbox{(}}n\neq 1{\mbox{)}}\,}
∫
x
cosh
a
x d
x
=
1
a
x
sinh
a
x
−
1 a 2
cosh
a
x
+
C {\displaystyle \int x\cosh ax\,dx={\frac {1}{a}}x\sinh ax-{\frac {1}{a^{2}}}\cosh ax+C\,}
∫ x 2
cosh
a
x d
x
=
− 2
x
cosh
a
x
a 2
+ (
x 2
a
+
2 a 3 ) sinh
a
x
+
C {\displaystyle \int x^{2}\cosh ax\,dx=-{\frac {2x\cosh ax}{a^{2}}}+\left({\frac {x^{2}}{a}}+{\frac {2}{a^{3}}}\right)\sinh ax+C\,}
∫
cosh
(
a
x
+
b
)
sin
(
c
x
+
d
) d
x
=
a
a 2
+ c 2 sinh
(
a
x
+
b
)
sin
(
c
x
+
d
)
−
c
a 2
+ c 2 cosh
(
a
x
+
b
)
cos
(
c
x
+
d
)
+
C {\displaystyle \int \cosh(ax+b)\sin(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\sinh(ax+b)\sin(cx+d)-{\frac {c}{a^{2}+c^{2}}}\cosh(ax+b)\cos(cx+d)+C\,}
∫
cosh
(
a
x
+
b
)
cos
(
c
x
+
d
) d
x
=
a
a 2
+ c 2 sinh
(
a
x
+
b
)
cos
(
c
x
+
d
)
+
c
a 2
+ c 2 cosh
(
a
x
+
b
)
sin
(
c
x
+
d
)
+
C {\displaystyle \int \cosh(ax+b)\cos(cx+d)\,dx={\frac {a}{a^{2}+c^{2}}}\sinh(ax+b)\cos(cx+d)+{\frac {c}{a^{2}+c^{2}}}\cosh(ax+b)\sin(cx+d)+C\,}
∫
cosh
a
x
cosh
b
x d
x
=
1
a 2
− b 2 (
a
sinh
a
x
cosh
b
x
−
b
sinh
b
x
cosh
a
x
)
+
C ( a 2
≠ b 2
) {\displaystyle \int \cosh ax\cosh bx\,dx={\frac {1}{a^{2}-b^{2}}}(a\sinh ax\cosh bx-b\sinh bx\cosh ax)+C\qquad {\mbox{(}}a^{2}\neq b^{2}{\mbox{)}}\,}
شرح مبسط
هذه قائمة تكاملات الدوال الزائدية.أخذا بالعلم أن
a
{\displaystyle a}
عدد غير منعدم وأن
C
{\displaystyle C}
هي ثابت التكامل.
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