Download Integral Calculus Formula Sheet

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
 ca
i
i 1
n
a
i
i 1
b
n
 c  ai

i 1
 bi   ai   bi x  b  a i 1
i 1
n
  height of ith rectangle    width of ith rectangle 
i
i 1
Right Endpoint Rule:
n(n  1)
i

2
i 1
n
 i2 
n
i
i 1
3
i 1
n
1  n
i 1
n 
a
n
n
n
n
f ( x)dx  lim  f (a  ix)x
n
n
i 1
i 1
 f (a  ix)(x)   (
n(n  1)(2n  1)
6
 n( n  1) 


 2 
2
(b  a )
n
) f (a  i (b n a ) )
Left Endpoint Rule:
n

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
b

Displacement: v ( x) dx 
Distance Traveled: v( x) dx a
a
t
t
0
0
s (t )  s (0)   v( x)dx Q (t )  Q (0)   Q( x)dx
Trig Formulas: sin x
1
sec x 
cos x
cos x
cos x
1
cos 2 ( x)  12 1  cos(2 x)  cot x 
csc x 
sin x
sin x
sin 2 ( x)  12 1  cos(2 x)  tan 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: A  s2 Area of a Triangle: A  12 bh Area of an Equilateral Trangle:
A
3
4
s2 Area of a Circle: A   r2 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
Washer Method: For volumes of revolution not laying on the axis with slices perpendicular to the axis c
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 (for 3‐D objects) Mass = Density * Area (for 2‐D objects) Mass = Density * Length (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: 1
u Inverse Trig Function 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 ) dv Exponential Functions ( sin
3
3x
3x
( e ,5 , 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): 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)  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. Trig Substitution: Expression Substitution a2  u2 u  a sin  
a2  u2 u  a tan  
u 2  a2 u  a sec  Domain 
2

2
Simplification 
  a 2  u 2  a cos   
a 2  u 2  a sec  2
 2
0     , 
 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.