eight

Differentiation:General Formulas
d
dx

c
0 d
dx
cf
x
cf
x
d
dx
f
x  g
x
f
x  g
x d
dx
f
x
g
x
f
x
g
x
d
dx
f
xg
x
f
xg
x  g
xf
x d
dx
f
x
g
x

g
xf

x
f
xg

x
g
x2
d
dx
f
g
x
f
g
xg
x d
dx

xn
nxn
1
Differentiation: Exponential and Logarithmic Functions
d
dx

ex
ex d
dx

ax
ax lna
d
dx ln|x|
1x
d
dx

logax
1
x lna
Differentiation: Trigonometric Functions
d
dx

sinx
cos x d
dx

cos x

sinx d
dx

tanx
sec2x
d
dx

csc x

csc x cot x d
dx

sec x
sec x tanx d
dx

cot x

csc2x
Differentiation: Inverse Trigonometric Functions
d
dx

sin
1x
1
1
x2
d
dx

cos
1x

1
1
x2
d
dx

tan
1x
1
1x2
d
dx

csc
1x

1
x x2
1
d
dx

sec
1x
1
x x2
1
d
dx

cot
1x

1
1x2
Integration Formulas
 udv
uv
 vdu  sec2u du
tanu  C  cscu du
ln|cscu
cotu|C
 undu
un1
n1
 C, n  1  csc2u du

cotu  C  1
a2
u2
du
sin
1 u
a
 C
 1u
du
ln|u|  C  secu tanu du
secu  C  1
a2u2 du
1
a tan
1 u
a
 C
 eudu
eu  C  cscu cotu

cscu  C  1
u u2
a2
du
1
a sec
1 u
a
 C
 audu
au
lna
 C  tanu du
ln|secu|C  1
a2
u2 du
1
2a ln| ua
u
a |  C
 sinu du

cos u  C  cotu du
ln|sinu|C  1
u2
a2 du
1
2a ln| u
a
ua |  C
 cos u du
sinu  C  secu du
ln|secu  tanu|C
Integration by Parts (order for choosing u)
Logarithmic Inverse Trig. Algebraic Trigonometric Exponential
Trigonometric Identities
1  tan2

sec2
sin
x  y
sinx cos y  cos x siny sin2x
2 sinx cos x
1  cot2

csc2
sin
x
y
sinx cos y
cos x siny cos 2x
cos2x
sin2x
tan
x  y
tanxtany
1
tanx tany cos
x  y
cos x cos y
sinx siny
2 cos2x
1
tan
x
y
tanx
tany
1tanx tany cos
x
y
cos x cos y  sinx siny
1
2 sin2x
Miscellaneous
Quad. form.: x

b b2
4ac
2a Vol. of sphere: V
4
3
r3
Eq. of circle:
x
h2 
y
k2
r2 Vol. of cylinder: V
r2h
logax
y  ay
x
x  y2
x2  2xy  y2
x2
y2

x  y
x
y
x
y2
x2
2xy  y2
x3  y3

x  y
x2
xy  y2
x  y3
x3  3x2y  3xy2  y3
x3
y3

x
y
x2  xy  y2
x
y3
x3
3x2y  3xy2
y3
Unit Circle