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:
Calculus Made Easy.pdf/273
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d
x
d
y
{\displaystyle {\frac {dx}{dy}}}
←
y
{\displaystyle y}
→
∫
y
d
x
{\displaystyle \int ydx}
Hyperbolic.
cosh
x
{\displaystyle \cosh x}
sinh
x
{\displaystyle \sinh x}
cosh
x
+
C
{\displaystyle \cosh x+C}
sinh
x
{\displaystyle \sinh x}
cosh
x
{\displaystyle \cosh x}
sinh
x
+
C
{\displaystyle \sinh x+C}
sech
2
x
{\displaystyle \operatorname {sech} ^{2}x}
tanh
x
{\displaystyle \tanh x}
log
ϵ
cosh
x
+
C
{\displaystyle \log _{\epsilon }\cosh x+C}
Miscellaneous.
−
1
(
x
+
a
)
2
{\displaystyle -{\frac {1}{(x+a)^{2}}}}
1
x
+
a
{\displaystyle {\frac {1}{x+a}}}
log
ϵ
(
x
+
a
)
+
C
{\displaystyle \log _{\epsilon }(x+a)+C}
−
x
(
a
2
+
x
2
)
3
2
{\displaystyle -{\frac {x}{(a^{2}+x^{2})^{\tfrac {3}{2}}}}}
1
a
2
+
x
2
{\displaystyle {\frac {1}{\sqrt {a^{2}+x^{2}}}}}
log
ϵ
(
x
+
a
2
+
x
2
)
+
C
{\displaystyle \log _{\epsilon }(x+{\sqrt {a^{2}+x^{2}}})+C}
∓
b
(
a
±
b
x
)
2
{\displaystyle \mp {\frac {b}{(a\pm bx)^{2}}}}
1
a
±
b
x
{\displaystyle {\frac {1}{a\pm bx}}}
±
1
b
log
ϵ
(
a
±
b
x
)
+
C
{\displaystyle \pm {\frac {1}{b}}\log _{\epsilon }(a\pm bx)+C}
−
3
a
x
x
(
a
2
+
x
2
)
5
2
{\displaystyle {\frac {-3a^{x}x}{(a^{2}+x^{2})^{\tfrac {5}{2}}}}}
a
2
(
a
2
+
x
2
)
3
2
{\displaystyle {\frac {a^{2}}{(a^{2}+x^{2})^{\tfrac {3}{2}}}}}
x
a
2
+
x
2
+
C
{\displaystyle {\frac {x}{\sqrt {a^{2}+x^{2}}}}+C}
a
⋅
cos
a
x
{\displaystyle a\cdot \cos ax}
sin
a
x
{\displaystyle \sin ax}
1
a
cos
a
x
+
C
{\displaystyle {\frac {1}{a}}\cos ax+C}
−
a
⋅
sin
a
x
{\displaystyle -a\cdot \sin ax}
cos
a
x
{\displaystyle \cos ax}
1
a
sin
a
x
+
C
{\displaystyle {\frac {1}{a}}\sin ax+C}
a
⋅
sec
2
a
x
{\displaystyle a\cdot \sec ^{2}ax}
tan
a
x
{\displaystyle \tan ax}
−
1
a
log
ϵ
cos
a
x
+
C
{\displaystyle -{\frac {1}{a}}\log _{\epsilon }\cos ax+C}
sin
2
x
{\displaystyle \sin 2x}
sin
2
x
{\displaystyle \sin ^{2}x}
x
2
−
sin
2
x
4
+
C
{\displaystyle {\frac {x}{2}}-{\frac {\sin 2x}{4}}+C}
−
sin
2
x
{\displaystyle -\sin 2x}
cos
2
x
{\displaystyle \cos ^{2}x}
x
2
+
sin
2
x
4
+
C
{\displaystyle {\frac {x}{2}}+{\frac {\sin 2x}{4}}+C}
n
⋅
sin
n
−
1
x
⋅
cos
x
{\displaystyle n\cdot \sin ^{n-1}x\cdot \cos x}
sin
n
x
{\displaystyle \sin ^{n}x}
cos
x
n
sin
n
−
1
x
+
n
−
1
n
∫
sin
n
−
2
x
d
x
+
C
{\displaystyle {\frac {\cos x}{n}}\sin ^{n-1}x+{\frac {n-1}{n}}\int \sin ^{n-2}xdx+C}
−
cos
x
sin
2
x
{\displaystyle -{\frac {\cos x}{\sin ^{2}x}}}
1
sin
x
{\displaystyle {\frac {1}{\sin x}}}
log
ϵ
tan
x
2
+
C
{\displaystyle \log _{\epsilon }\tan {\frac {x}{2}}+C}
−
sin
2
x
sin
4
x
{\displaystyle -{\frac {\sin 2x}{\sin ^{4}x}}}
1
sin
2
x
{\displaystyle {\frac {1}{\sin ^{2}x}}}
−
cotan
x
+
C
{\displaystyle -\operatorname {cotan} x+C}
sin
2
x
−
cos
2
x
sin
2
x
⋅
cos
2
x
{\displaystyle {\frac {\sin ^{2}x-\cos ^{2}x}{\sin ^{2}x\cdot \cos ^{2}x}}}
1
sin
x
⋅
cos
x
{\displaystyle {\frac {1}{\sin x\cdot \cos x}}}
log
ϵ
tan
x
+
C
{\displaystyle \log _{\epsilon }\tan x+C}
n
⋅
sin
m
x
⋅
cos
n
x
+
m
⋅
sin
n
x
⋅
cos
m
x
{\displaystyle n\cdot \sin mx\cdot \cos nx+m\cdot \sin nx\cdot \cos mx}
sin
m
x
⋅
sin
n
x
{\displaystyle \sin mx\cdot \sin nx}
1
2
cos
(
m
−
n
)
x
−
1
2
cos
(
m
+
n
)
x
+
C
{\displaystyle {\tfrac {1}{2}}\cos(m-n)x-{\tfrac {1}{2}}\cos(m+n)x+C}
2
a
⋅
sin
2
a
x
{\displaystyle 2a\cdot \sin 2ax}
sin
2
a
x
{\displaystyle \sin ^{2}ax}
x
2
−
sin
2
a
x
4
a
+
C
{\displaystyle {\frac {x}{2}}-{\frac {\sin 2ax}{4a}}+C}
−
2
a
⋅
sin
2
a
x
{\displaystyle -2a\cdot \sin 2ax}
cos
2
a
x
{\displaystyle \cos ^{2}ax}
x
2
+
sin
2
a
x
4
a
+
C
{\displaystyle {\frac {x}{2}}+{\frac {\sin 2ax}{4a}}+C}
→
