This document provides formulas and rules for calculus, including: - Derivative rules for common functions like sin, cos, ln, and e^x. - Properties of integrals, such as linearity and the Fundamental Theorem of Calculus. - Formulas for area, volume, work, force, and other physical applications that use calculus. - Guidelines for integration by parts and strategies for integrals involving trigonometric functions.

1. Integral Calculus Formula Sheet
Derivative Rules:
  0
d
c
dx
   1n nd
x nx
dx


 sin cos
d
x x
dx

 sec sec tan
d
x x x
dx

  2
tan sec
d
x x
dx

 cos sin
d
x x
dx
 
 csc csc cot
d
x x x
dx
 
  2
cot csc
d
x x
dx
 
  lnx xd
a a a
dx

 x xd
e e
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       2
f g fgf
g g
    
 
 
         
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    
( ) 0
a
a
f x dx  ( ) ( )
b a
a b
f x dx f x dx  
( ) ( ) ( )
c b c
a a b
f x dx f x dx f x dx   
1
( )
b
ave
a
f f x dx
b a

 
0
( ) 2 ( )
a a
a
f x dx f x dx

  if f(x) is even ( ) 0
a
a
f x dx

 if f(x) is odd
( )
( )
( ( )) ( ) ( )
f bb
a f a
g f x f x dx g u du   udv uv vdu  
Integration Rules:
du u C 
1
1
n
n u
u du C
n

 

ln
du
u C
u
 
u u
e du e C 
1
ln
u u
a du a C
a
 
sin cosu du u C  
cos sinu du u C 
2
sec tanu du u C 
2
csc cotu u C  
csc cot cscu u du u C  
sec tan secu u du u C 
2 2
1
arctan
du u
C
a u a a
 
  
  

2 2
arcsin
du u
C
aa u
 
  
 

2 2
1
sec
udu
arc C
a au u a
 
  
  


2. Fundamental Theorem of Calculus:
     '  
x
a
d
F x f t dt f x
dx
where  f t is a continuous function on [a, x].
      
b
a
f x dx F b F a , where F(x) is any antiderivative of f(x).
Riemann Sums:
1 1
n n
i i
i i
ca c a
 
 
1 1 1
n n n
i i i i
i i i
a b a b
  
    
1
( ) lim ( )
b n
n
ia
f x dx f a i x x


   
n
ab
x


1
1
n
i
n


1
( 1)
2
n
i
n n
i



2
1
( 1)(2 1)
6
n
i
n n n
i

 

2
3
1
( 1)
2
n
i
n n
i

 
  
 

   height of th rectangle width of th rectangle
i
i i
Right Endpoint Rule:
 



n
i
n
ab
n
ab
n
i
iafxxiaf
1
)()(
1
)()()()(
Left Endpoint Rule:
( ) ( )
1 1
( ( 1) )( ) ( ) ( ( 1) )
n n
b a b a
n n
i i
f a i x x f a i 
 
       
Midpoint Rule:
   ( 1) ( ) ( 1) ( )
2 2
1 1
( )( ) ( ) ( )
n n
i i b a i i b a
n n
i i
f a x x f a     
 
     
Net Change:
Displacement: ( )
b
a
v x dx Distance Traveled: ( )
b
a
v x dx
0
( ) (0) ( )
t
s t s v x dx  
0
( ) (0) ( )
t
Q t Q Q x dx  
Trig Formulas:
 2 1
2sin ( ) 1 cos(2 )x x 
sin
tan
cos
x
x
x

1
sec
cos
x
x

cos( ) cos( )x x  2 2
sin ( ) cos ( ) 1x x 
 2 1
2cos ( ) 1 cos(2 )x x 
cos
cot
sin
x
x
x

1
csc
sin
x
x

sin( ) sin( )x x   2 2
tan ( ) 1 sec ( )x x 
Geometry Fomulas:
Area of a Square:
2
A s
Area of a Triangle:
1
2A bh
Area of an
Equilateral Trangle:
23
4A s
Area of a Circle:
2
A r
Area of a
Rectangle:
A bh

3. Areas and Volumes:
Area in terms of x (vertical rectangles):
( )
b
a
top bottom dx
Area in terms of y (horizontal rectangles):
( )
d
c
right left dy
General Volumes by Slicing:
Given: Base and shape of Cross‐sections
( )
b
a
V A x dx  if slices are vertical
( )
d
c
V A y dy  if slices are horizontal
Disk Method:
For volumes of revolution laying on the axis with
slices perpendicular to the axis
 
2
( )
b
a
V R x dx  if slices are vertical
 
2
( )
d
c
V R y dy  if slices are horizontal
Washer Method:
For volumes of revolution not laying on the axis with
slices perpendicular to the axis
   
2 2
( ) ( )
b
a
V R x r x dx   if slices are vertical
   
2 2
( ) ( )
d
c
V R y r y dy   if slices are horizontal
Shell Method:
For volumes of revolution with slices parallel to the
axis
2
b
a
V rhdx  if slices are vertical
2
d
c
V rhdy  if slices are horizontal
Physical Applications:
Physics Formulas Associated Calculus Problems
Mass:
Mass = Density * Volume (for 3‐D objects)
Mass = Density * Area (for 2‐D objects)
Mass = Density * Length (for 1‐D objects)
Mass of a one‐dimensional object with variable linear
density:
( ) ( )
b b
distancea a
Mass linear density dx x dx  
Work:
Work = Force * Distance
Work = Mass * Gravity * Distance
Work = Volume * Density * Gravity * Distance
Work to stretch or compress a spring (force varies):

'
( ) ( )
b b b
Hooke s Lawa a a
for springs
Work force dx F x dx kx dx    
Work to lift liquid:
( )( )( )( )
9.8* * ( )*( ) ( )
d
c volume
d
c
Work gravity density distance areaof a slice dy
W A y a y dy inmetric

 



Force/Pressure:
Force = Pressure * Area
Pressure = Density * Gravity * Depth
Force of water pressure on a vertical surface:
( )( )( )( )
9.8* *( )* ( ) ( )
d
c area
d
c
Force gravity density depth width dy
F a y w y dy inmetric

 




4. Integration by Parts:
Knowing which function to call u and which to call dv takes some practice. Here is a general guide:
u Inverse Trig Function (
1
sin ,arccos ,x x
etc )
Logarithmic Functions ( log3 ,ln( 1),x x  etc )
Algebraic Functions (
3
, 5,1/ ,x x x etc)
Trig Functions (sin(5 ),tan( ),x x etc )
dv Exponential Functions (
3 3
,5 ,x x
e etc )
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): Integrals involving sec(x) and tan(x):
1. If the power of the sine is odd and positive:
Goal: cosu x
i. Save a sin( )du x dx
ii. Convert the remaining factors to
cos( )x (using 2 2
sin 1 cosx x  .)
1. If the power of sec( )x is even and positive:
Goal: tanu x
i. Save a
2
sec ( )du x dx
ii. Convert the remaining factors to
tan( )x (using
2 2
sec 1 tanx x  .)
2. If the power of the cosine is odd and positive:
Goal: sinu x
i. Save a cos( )du x dx
ii. Convert the remaining factors to
sin( )x (using 2 2
cos 1 sinx x  .)
2. If the power of tan( )x is odd and positive:
Goal: sec( )u x
i. Save a sec( ) tan( )du x x dx
ii. Convert the remaining factors to
sec( )x (using
2 2
sec 1 tanx x  .)
3. If both sin( )x and cos( )x have even powers:
Use the half angle identities:
i.  2 1
2
sin ( ) 1 cos(2 )x x 
ii.  2 1
2
cos ( ) 1 cos(2 )x x 
 If there are no sec(x) factors and the power of
tan(x) is even and positive, use
2 2
sec 1 tanx x 
to convert one
2
tan x to
2
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.
Trig Substitution:
Expression Substitution Domain Simplification
2 2
a u sinu a 
2 2
 
   2 2
cosa u a  
2 2
a u tanu a 
2 2
 
   2 2
seca u a  
2 2
u a secu a  0 ,
2

     2 2
tanu a a  
Partial Fractions:
Linear factors: Irreducible quadratic factors:
2 1
1 1 1 1 1
( )
...
( ) ( ) ( ) ( ) ( )m m m
P x A B Y Z
x r x r x r x r x r
    
    
2 2 2 2 2 1 2
1 1 1 1 1
( )
...
( ) ( ) ( ) ( ) ( )m m m
P x Ax B Cx D Wx X Yx Z
x r x r x r x r x r
   
    
    
If the fraction has multiple factors in the denominator, we just add the decompositions.