Download Integration Formulas

Math 180
Derivatives and Integrals
Basic Differentiation Rules
d
cu   cu '
dx
d  u  vu'uv '

4.
dx  v 
v2
d
x   1
7.
dx
d u
e  e u u'
10.
dx
d
sin u   (cos u ) u '
13.
dx
16.
d
csc u   ( csc u cot u ) u '
dx
d
19.
arcsin u  u' 2
dx
1 u
1.
 
d
u  v  u '  v'
dx
d
c  0
5.
dx
d
8.
u   u u '
dx
u
2.
 
 
d u
a  a u (ln a)u '
dx
d
cos u   ( sin u ) u '
14.
dx
d
sec u   (sec u tan u ) u '
17.
dx
11.
20.
d
uv  uv'  vu'
dx
d n
u  nu n 1u '
6.
dx
d
ln u   1  u '  u '
9.
dx
u
u
d
12.
log a u   1  u'  u'
dx
u (ln a)
u (ln a)
d
tan u   (sec 2 u ) u '
15.
dx
d
cot u   ( csc 2 u ) u '
18.
dx
3.
d
arccos u    u' 2
dx
1 u
21.
d
arctan u   u ' 2
dx
1 u
Basic Integration Rules
1.
3.
 k f u du  k  f u du
 du  u  C
u n1
 C ; n  1
n 1
7.  e u du  e u  C
5.
n
 u du 
 sin u du   cos u  C
11.  tan u du   ln cos u  C
13.  csc u du   ln csc u  cot u  C
15.  sec u du  tan u  C
17.  sec u tan u du  sec u  C
9.
2
19.
a
2
du
1
u
 arctan  C
2
a
a
u
2.
4.
 [ f u   g u ] du   f u du   g u du
 k  du  k u  C
du
 ln u  C
u
1
au
C 
C
8.  a u du  a u 
ln a
ln a
10.  cos u du  sin u  C
6.
1
 u du  
12.  cot u du  ln sin u  C
14.  sec u du  ln sec u  tan u  C
16.  csc 2 u du   cot u  C
18.  csc u cot u du   csc u  C
20.

du
a2  u2
 arcsin
u
C
a
Integration by Parts:
 u dv  uv   v du
Arc Length:
b
S   1   f ' ( x)  dx
2
a
Math 190 Formula Sheet
Powers of the Trig Functions:
1) Integrals of the form:  sin m x cos n x dx m or n odd
Strategy: If m is odd, save a sine factor and convert to cosine
If n is odd, save a cosine factor and convert to sine
2) Integrals of the form:
 sin
m
x cos n x dx m and n even and non-negative
Strategy: use ½ angle identities:
1  cos( 2 x)
1  cos( 2 x)
cos 2 x 
; sin 2 x 
2
2
3) Integrals of the form:  sec m x tan n x dx ; if m is even
Strategy: Save a sec 2 x and convert to tangent
4) Integrals of the form:  sec m x tan n x dx ; if n is odd
Strategy: Save a secx tanx and convert to secant
5)  tan n x dx n any positive integer
Strategy: convert a tan2 x to sec2x-1 and distribute; repeat if necessary
6)  sec m x dx ; m is odd
Strategy: Integrate by parts
*** If all else fails convert everything to sines and cosines
Trigonometric Substitution:
Form
Trig Sub
Identity
x  a tan 
1  tan 2   sec 2 
a2  x2
a2  x2
x2  a2
x  a sin 
1  sin 2   cos 2 
x  a sec
sec 2   1  tan 2 
Trigonometric Identities:
sin 2 x  cos 2 x  1
sec 2 x  1  tan 2 x
csc 2 x  1  cot 2 x
1
tan x
1
sec x 
cos x
1
csc x 
sin x
cot x 
sin 2x  2 sin x cos x
cos 2 x  cos 2 x  sin 2 x
Numerical Integration Approximations:
ba
 f x0   2 f x1   2 f x2   ...  2 f xn1   f xn 
TRAP (n) 
2n
ba
 f x0   4 f x1   2 f x2   4 f x3   ...  2 f xn2   4 f xn1   f xn 
SIMP (n) 
3n