Differentiation Formulas
d
k=0
dx
d
[f (x) ± g(x)] = f 0 (x) ± g 0 (x)
dx
d
[k · f (x)] = k · f 0 (x)
dx
d
[f (x)g(x)] = f (x)g 0 (x) + g(x)f 0 (x)
dx
g(x)f 0 (x) − f (x)g 0 (x)
d f (x)
=
2
dx g(x)
[g(x)]
d
f (g(x)) = f 0 (g(x)) · g 0 (x)
dx
d n
x = nxn−1
dx
d
sin x = cos x
dx
d
cos x = − sin x
dx
d
tan x = sec2 x
dx
d
cot x = − csc2 x
dx
d
sec x = sec x tan x
dx
d
csc x = − csc x cot x
dx
d x
e = ex
dx
d x
a = ax ln a
dx
1
d
ln |x| =
dx
x
d
1
sin−1 x = √
dx
1 − x2
d
−1
cos−1 x = √
dx
1 − x2
d
1
tan−1 x = 2
dx
x +1
d
−1
cot−1 x = 2
dx
x +1
d
1
√
sec−1 x =
dx
|x| x2 − 1
d
−1
√
csc−1 x =
dx
|x| x2 − 1
Integration Formulas
Z
(1)
dx = x + C
(2)
Z
(3)
Z
xn dx =
xn+1
+C
n+1
(1)
(2)
dx
= ln |x| + C
x
(3)
ex dx = ex + C
(4)
(4)
Z
(5)
Z
ax dx =
(6)
1 x
a +C
ln a
(5)
Z
(7)
(8)
ln x dx = x ln x − x + C
(6)
sin x dx = − cos x + C
(7)
cos x dx = sin x + C
(8)
tan x dx = − ln | cos x| + C
(9)
Z
(9)
Z
(10)
Z
(11)
Z
cot x dx = ln | sin x| + C
(10)
sec x dx = ln | sec x + tan x| + C
(11)
(12)
Z
(13)
Z
csc x dx = − ln | csc x + cot x| + C (12)
(14)
(15)
Z
(16)
Z
sec2 x dx = tan x + C
(13)
csc2 x dx = − cot x + C
(14)
sec x tan x dx = sec x + C
(15)
csc x cot x dx = − csc x + C
(16)
x
dx
= sin−1 + C
a
a2 − x2
(17)
dx
1
x
= tan−1 + C
a2 + x2
a
a
(18)
dx
1
|x|
√
= sec−1
+C
2
2
a
a
x x −a
(19)
(17)
Z
(18)
Z
(19)
Z
(20)
Z
(21)
(22)
Z
√
Integral Calculus Formula Sheet
Derivative Rules:
d
c   0
dx
d
 x n   nx n 1
dx
d
 sin x   cos x
dx
d
 sec x   sec x tan x
dx
d
 tan x   sec2 x
dx
d
 cos x    sin x
dx
d
 csc x    csc x cot x
dx
d
 cot x    csc 2 x
dx
d x
a   a x ln a

dx
d x
e   ex
dx
d
d
cf  x    c f  x  
dx
dx
d
d
d
f  x   g  x    f  x     g  x  
dx
dx
dx
f  g   f   g  f  g 
f

g
  fg 
 f g
 
g2

d
f g x   f  g x  g  x 
dx
 



Properties of Integrals:
 kf (u )du  k  f (u )du
  f (u )  g (u )du   f (u )du   g (u )du
a
b
 f ( x)dx  0
 f ( x)dx    f ( x)dx
a
c
a
b
a
a

b
f ave 
b
a
1
f ( x) dx
b  a a
a

f ( x)dx  2 f ( x) dx if f(x) is even
a
b
c
 f ( x)dx   f ( x)dx   f ( x)dx
a
a

f ( x) dx  0 if f(x) is odd
a
0
b
f (b )
a
f (a)
 g ( f ( x)) f ( x)dx  
 udv  uv   vdu
g (u )du
Integration Rules:
 du  u  C
n 1
u
 u du  n  1  C
du
 u  ln u  C
u
u
 e du  e  C
n
1
 a du  ln a a
u
u
C
 sin u du   cos u  C
 cos u du  sin u  C
 sec u du  tan u  C
 csc u   cot u  C
 csc u cot u du   csc u  C
 sec u tan u du  sec u  C
2
2
du
1
u
 arctan    C
2
a
u
a
du
u
 a 2  u 2  arcsin  a   C
u
1
du
 u u 2  a 2  a arc sec  a   C
a
2
Fundamental Theorem of Calculus:
F ' x 
d x
f  t  dt  f  x  where f  t  is a continuous function on [a, x].
dx  a
 f  x  dx  F  b   F  a  , where F(x) is any antiderivative of f(x).
b
a
Riemann Sums:
n
 c  ai
i
i 1
n
i 1
ba
n
  height of ith rectangle    width of ith rectangle 
i 1
1  n
i
i 1
Right Endpoint Rule:
n(n  1)
i

2
i 1
n
 i2 
i 1
n
i
i 1
3
i 1
x 
n
n
n 
a
n
 bi   ai   bi
i
n
f ( x)dx  lim  f (a  ix)x

i 1
n
a
i 1
b
n
 ca
n
n
i 1
i 1
 f (a  ix)(x)   (
n(n  1)(2n  1)
6
 n( n  1) 


 2 
(b  a )
n
) f (a  i (b n a ) )
Left Endpoint Rule:
n

2
i 1
n
f (a  (i  1)x)(x)   ( (b na ) ) f (a  (i  1) (bn a ) )
i 1
Midpoint Rule:
n

f (a 
i 1

( i 1)  i
2

n
x)(x)   ( (b n a ) ) f (a 
i 1

( i 1)  i
2

(ba )
n
)
Net Change:
b
Displacement:
 v( x)dx
b
Distance Traveled:
a

t
v( x) dx
s (t )  s (0)   v( x)dx
0
a
t
Q (t )  Q (0)   Q( x)dx
0
Trig Formulas:
sin x
cos x
cos x
cot x 
sin x
sin 2 ( x)  12 1  cos(2 x) 
tan x 
cos 2 ( x)  12 1  cos(2 x) 
1
cos x
1
csc x 
sin x
sec x 
cos(  x )  cos( x )
sin 2 ( x)  cos 2 ( x)  1
sin( x )   sin( x )
tan 2 ( x)  1  sec 2 ( x)
Geometry Fomulas:
Area of a Square:
Area of a Triangle:
As
A  bh
2
1
2
Area of an
Equilateral Trangle:
A
3
4
s
2
Area of a Circle:
A r
2
Area of a
Rectangle:
A  bh
Areas and Volumes:
Area in terms of y (horizontal rectangles):
Area in terms of x (vertical rectangles):
b
d
 (top  bottom)dx
 (right  left )dy
General Volumes by Slicing:
Given: Base and shape of Cross‐sections
Disk Method:
For volumes of revolution laying on the axis with
slices perpendicular to the axis
a
c
b
V   A( x )dx if slices are vertical
b
V     R ( x )  dx if slices are vertical
2
a
d
a
V   A( y )dy if slices are horizontal
d
V     R ( y )  dy if slices are horizontal
2
c
c
Washer Method:
For volumes of revolution not laying on the axis with
slices perpendicular to the axis
Shell Method:
For volumes of revolution with slices parallel to the
axis
b
b
V     R ( x)     r ( x)  dx if slices are vertical
V   2 rhdx if slices are vertical
V     R ( y )     r ( y )  dy if slices are horizontal
V   2 rhdy if slices are horizontal
2
2
a
d
2
2
c
a
d
c
Physical Applications:
Physics Formulas
Mass:
Mass = Density * Volume
Mass = Density * Area
Mass = Density * Length
(for 3‐D objects)
(for 2‐D objects)
(for 1‐D objects)
Associated Calculus Problems
Mass of a one‐dimensional object with variable linear
density:
b
b
Mass   (linear density ) dx
    ( x)dx
distance
a
Work:
Work = Force * Distance
Work = Mass * Gravity * Distance
Work = Volume * Density * Gravity * Distance
a
Work to stretch or compress a spring (force varies):
b
b
b
a
a Hooke ' s Law
for springs
Work   ( force)dx   F ( x)dx  
a
kx

dx
Work to lift liquid:
d
Work   ( gravity )(density )(distance) ( area of a slice) dy



c
volume
d
W   9.8*  * A( y ) *(a  y )dy (in metric)
c
Force/Pressure:
Force = Pressure * Area
Pressure = Density * Gravity * Depth
Force of water pressure on a vertical surface:
d
Force   ( gravity )( density )(depth) ( width) dy


c
d
area
F   9.8*  *(a  y ) * w( y )dy (in metric)
c
Integration by Parts:
Knowing which function to call u and which to call dv takes some practice. Here is a general guide:
u
dv
1
Logarithmic Functions
x, arccos x, etc )
( log 3 x, ln( x  1), etc )
Algebraic Functions
( x , x  5,1/ x, etc )
Trig Functions
( sin(5 x ), tan( x ), etc )
Exponential Functions
( e ,5 , etc )
Inverse Trig Function
( sin
3
3x
3x
Functions that appear at the top of the list are more like to be u, functions at the bottom of the list are more like to be dv.
Trig Integrals:
Integrals involving sin(x) and cos(x):
1.
2.
3.
Integrals involving sec(x) and tan(x):
If the power of the sine is odd and positive:
Goal: u  cos x
i. Save a du  sin( x ) dx
ii. Convert the remaining factors to
cos( x ) (using sin 2 x  1  cos 2 x .)
1.
If the power of the cosine is odd and positive:
Goal: u  sin x
i. Save a du  cos( x ) dx
ii. Convert the remaining factors to
sin( x ) (using cos 2 x  1  sin 2 x .)
2.
If both sin( x ) and cos( x ) have even powers:
Use the half angle identities:
i.
sin ( x ) 
1
ii.
cos ( x ) 
1
2
2
2
2
1  cos(2 x ) 
1  cos(2 x) 
Trig Substitution:
Expression
If the power of sec( x ) is even and positive:
Goal: u  tan x
i. Save a du  sec ( x ) dx
ii. Convert the remaining factors to
2
tan( x ) (using sec x  1  tan x .)
2
2
If the power of tan( x ) is odd and positive:
Goal: u  sec( x )
i. Save a du  sec( x ) tan( x ) dx
ii. Convert the remaining factors to
sec( x ) (using sec x  1  tan x .)
2

2
If there are no sec(x) factors and the power of
tan(x) is even and positive, use sec x  1  tan x
2
2
2
2
to convert one tan x to sec x
 Rules for sec(x) and tan(x) also work for csc(x) and
cot(x) with appropriate negative signs
If nothing else works, convert everything to sines and cosines.
Substitution
Domain
a2  u2
u  a sin 

a2  u2
u  a tan 

u 2  a2
u  a sec 

2

2
 
 
Simplification

a 2  u 2  a cos 
2

2
0     , 
a 2  u 2  a sec 

2
u 2  a 2  a tan 
Partial Fractions:
Linear factors:
Irreducible quadratic factors:
P( x)
A
B
Y
Z


 ... 

m
2
m 1
( x  r1 )
( x  r1 ) ( x  r1 )
( x  r1 )
( x  r1 ) m
P( x)
Ax  B
Cx  D
Wx  X
Yx  Z
 2
 2
 ...  2
 2
2
m
2
m 1
( x  r1 )
( x  r1 ) ( x  r1 )
( x  r1 )
( x  r1 ) m
If the fraction has multiple factors in the denominator, we just add the decompositions.