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
ba
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 ix)x
i 1
n
a
i 1
b
n
ca
n
n
i 1
i 1
f (a ix)(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) (bn 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
(ba )
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:
As
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.
0
