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Calculus Made Easy.pdf/272
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Table of standard forms
d
x
d
y
{\displaystyle {\frac {dx}{dy}}}
←
y
{\displaystyle y}
→
∫
y
d
x
{\displaystyle \int ydx}
Algebraic.
1
{\displaystyle 1}
x
{\displaystyle x}
1
2
x
2
+
C
{\displaystyle {\tfrac {1}{2}}x^{2}+C}
0
{\displaystyle 0}
a
{\displaystyle a}
a
x
+
C
{\displaystyle ax+C}
1
{\displaystyle 1}
x
±
a
{\displaystyle x\pm a}
1
2
x
2
±
a
x
+
C
{\displaystyle {\tfrac {1}{2}}x^{2}\pm ax+C}
a
{\displaystyle a}
a
x
{\displaystyle ax}
1
2
a
x
2
+
C
{\displaystyle {\tfrac {1}{2}}ax^{2}+C}
2
x
{\displaystyle 2x}
x
2
{\displaystyle x^{2}}
1
3
x
3
+
C
{\displaystyle {\tfrac {1}{3}}x^{3}+C}
n
x
n
−
1
{\displaystyle nx^{n-1}}
x
n
{\displaystyle x^{n}}
1
n
+
1
x
n
+
1
+
C
{\displaystyle {\frac {1}{n+1}}x^{n+1}+C}
−
x
−
2
{\displaystyle -x^{-2}}
x
−
1
{\displaystyle x^{-1}}
log
ϵ
x
+
C
{\displaystyle \log _{\epsilon }{x}+C}
d
u
d
x
±
d
v
d
x
±
d
w
d
x
{\displaystyle {\frac {du}{dx}}\pm {\frac {dv}{dx}}\pm {\frac {dw}{dx}}}
u
±
v
±
w
{\displaystyle u\pm v\pm w}
∫
u
d
x
±
∫
v
d
x
±
∫
w
d
x
{\displaystyle \int udx\pm \int vdx\pm \int wdx}
u
d
v
d
x
+
v
d
u
d
x
{\displaystyle u{\frac {dv}{dx}}+v{\frac {du}{dx}}}
u
v
{\displaystyle uv}
No general form known
v
d
u
d
x
−
u
d
v
d
x
v
2
{\displaystyle {\frac {v{\frac {du}{dx}}-u{\frac {dv}{dx}}}{v^{2}}}}
u
v
{\displaystyle {\frac {u}{v}}}
No general form known
d
u
d
x
{\displaystyle {\frac {du}{dx}}}
u
{\displaystyle u}
u
x
−
∫
x
d
u
+
C
{\displaystyle ux-\int xdu+C}
Exponential and Logarithmic.
ϵ
x
{\displaystyle \epsilon ^{x}}
ϵ
x
{\displaystyle \epsilon ^{x}}
ϵ
x
+
C
{\displaystyle \epsilon ^{x}+C}
x
−
1
{\displaystyle x^{-1}}
log
ϵ
x
{\displaystyle \log _{\epsilon }x}
x
(
log
ϵ
x
−
1
)
+
C
{\displaystyle x(\log _{\epsilon }x-1)+C}
0
⋅
4343
×
x
−
1
{\displaystyle 0\cdot 4343\times x^{-1}}
log
10
x
{\displaystyle \log _{10}x}
0
⋅
4343
x
(
log
ϵ
x
−
1
)
+
C
{\displaystyle 0\cdot 4343x(\log _{\epsilon }x-1)+C}
a
x
log
ϵ
a
{\displaystyle a^{x}\log _{\epsilon }a}
a
x
{\displaystyle a^{x}}
a
x
log
ϵ
a
+
C
{\displaystyle {\frac {a^{x}}{\log _{\epsilon }a}}+C}
Trigonometrical.
cos
x
{\displaystyle \cos x}
sin
x
{\displaystyle \sin x}
−
cos
x
+
C
{\displaystyle -\cos x+C}
−
sin
x
{\displaystyle -\sin x}
cos
x
{\displaystyle \cos x}
sin
x
+
C
{\displaystyle \sin x+C}
sec
2
x
{\displaystyle \sec ^{2}x}
tan
x
{\displaystyle \tan x}
−
log
ϵ
cos
x
+
C
{\displaystyle -\log _{\epsilon }\cos x+C}
Circular (Inverse).
1
(
1
−
x
2
)
{\displaystyle {\frac {1}{\sqrt {(1-x^{2})}}}}
arcsin
x
{\displaystyle \arcsin x}
x
⋅
arcsin
x
+
1
−
x
2
+
C
{\displaystyle x\cdot \arcsin x+{\sqrt {1-x^{2}}}+C}
−
1
(
1
−
x
2
)
{\displaystyle -{\frac {1}{\sqrt {(1-x^{2})}}}}
arccos
x
{\displaystyle \arccos x}
x
⋅
arccos
x
−
1
−
x
2
+
C
{\displaystyle x\cdot \arccos x-{\sqrt {1-x^{2}}}+C}
1
1
+
x
2
{\displaystyle {\frac {1}{1+x^{2}}}}
arctan
x
{\displaystyle \arctan x}
x
⋅
arctan
x
−
1
2
log
ϵ
(
1
+
x
2
)
+
C
{\displaystyle x\cdot \arctan x-{\tfrac {1}{2}}\log _{\epsilon }(1+x^{2})+C}