Calculus Rules

Trigonometric Functions
Differentiation Integration Important rules
d
sin u  = (cos u ) u '  sin x dx = − cos x + C sin 2 x + cos2 x = 1
dx
d
cosu  = (− sin u ) u '  cos x dx = sin x + C
dx sin 2 x = 2 sin x cos x
cos x = cos  x − sin  x
d
tan u  = (sec 2 u ) u '  tan x dx = ln sec x +C =  cos  x −
dx
= −  sin  x

d
cot u  = (− csc2 u ) u '  cot x dx = ln sin x +C sin  x = (− cos x )
dx 

d
sec u  = (sec u tan u ) u '  sec x dx = ln sec x + tan x +C cos  x = (+ cos x )
dx 
sin x cos y =
d
cscu  = (− cscu cot u) u '  csc x dx = ln csc x − cot x +C 
[sin(x − y ) + sin(x + y )]
dx

sin x sin y =

 sec 

x dx = tan x + C [cos(x − y ) − cos( x + y )]

cos x cos y =

 csc 

x dx = − cot x + C [cos(x − y ) + cos(x + y )]

tan x + tan y
tan(x + y ) =
− tan x tan y

Walls rules
  
( n − )( n − )..........   
 sin x  
 n
if even
  n dx =
cos x    


(n )(n − )..........    if odd

    
(n −)(n − )..........   .(m −)(m − )..........    
   

if both even
 sin n x .cos m x dx =
 

 

(n + m )(n + m − )..........   otherwise


pg. 1
 Hyperbolic Functions
Differentiation Integration Important rules
d
sh u  = (ch u )u '  sh x dx = ch x + C ch  x − sh  x =
dx
d
ch u  = (sh u )u '  ch x dx = sh x + C sh x = sh x ch x
dx
ch x = ch  x + sh  x
d
 tanh u  = (sech  u )u '  tanh x dx = ln ch x +C = ch  x −
dx
= + sh  x
d 
coth u  = (− csch  u )u '  coth x dx = ln sh x +C sh  x = (ch x −)
dx 
d 
sech u  = (− sech u tanh u )u '  sech x dx = tan
−
(sh x ) + C c h  x = (+ ch x )
dx 
x e x + e −x
 csch x dx = ln tanh 
d
csch u  = (− csch u coth u )u ' +C ch x =
dx 2
e − e −x
x
sh x =
2

basic Functions
Differentiation Integration Important rules
d f (x ) f (x )
dx
ln f (x )  =
f (x )  f (x ) dx = ln f (x ) + c ln x a = a ln x
d f (x ) f (x ) ln x
dx
loga f (x )  =
ln a f (x )  f (x ) dx =  f (x ) + c loga x =
ln a
d
e f  =ef f (x )
e f (x ) dx = e f ( x ) + C
(x ) (x ) f (x )
dx   ln xy = ln x + ln y
d ef (x ) x
= ln x − ln y
a f  = af f (x ) ln a a f (x ) dx = +C ln
(x ) (x ) f (x )

dx   ln a y
d
f (x )n = n f (x )n − f (x ) f (x )
n +

 f (x ) f (x ) dx =
n
+c y = ln x → x = e y
dx n +

pg. 2
 Inverse Trigonometric Functions
Differentiation Integration Important rules
d u' 1
sin − u  =
  csc−1 ( x ) = sin −1  
dx −u  x 
f (x ) dx f (x )
d −u '  = sin −1 +C
1
 cos − u  =
  a 2 − f 2 (x ) a s ec −1 ( x ) = cos −1  
dx −u  x 
1
cot −1 ( x ) = tan −1  
d u'
 tan − u  =
dx   + u  x 
f (x ) dx 1 f (x )
d
cot − u  =
−u ' a 2
= tan −1
+ f (x ) a
2
a
+C
dx   + u 

d u'
sec − u  =
 
dx u u  −
f (x ) dx 1 f (x )
d
csc− u  =
−u '  f (x ) = sec −1
f 2 (x ) − a 2 a a
+C
 
dx u u  −

pg. 3
 Inverse hyperbolic Functions
Differentiation Integration
f (x ) dx f (x )

d u'
sh − u  =
 
= sh −1 +C
dx +u  a 2 + f 2 (x ) a
f (x ) dx f (x )

d u'
ch − u  =
 
= ch −1 +C
dx u  − f 2 (x ) − a 2 a

d
th − u  =
u' 1 f (x )
   tanh −1 + C if f (x )  1
dx − u  f (x ) dx a a
 a 2 − f 2 (x ) =  1 −1 f (x )
d
cot − u  =
u'  coth + C if f (x )  1
dx   − u   a a
d −u ' f (x ) dx 1 f (x )
dx
sech − u  =
 
u − u 
 f (x ) a 2 − f 2 (x )
= − sech −1
a a
+C

d −u ' f (x ) dx 1 f (x )
dx
csch − u  =
 
u + u 
 f (x ) a + f (x )
2 2
= − csch −1
a a
+C

Important rules Logarithmic form

sech −1x = cosh −1

1
x ( )
sinh −1 x = ln x + x 2 + 1 , −   x  

csch −1x = sinh −1

1
x (
cosh −1 x = ln x + x 2 − 1 , x  1)
1 1 1+ x
coth −1x = tanh −1 tanh −1 x = ln , x 1
x 2 1− x
 1+ 1− x 2 
sech −1x = ln   , 0  x 1
 x 
 
1 1+ x 2 
csch −1x = ln  +  , x 0
x x 
 
1 x +1
coth −1 x = ln , x 1
2 x −1

pg. 4
 Trigonometric rules
 sin
n
1. x cos m x dx

Case Solution
 sin
n −1
n odd (+ve) integer x cos x (sin x dx ), set every sin 2 x = 1 − cos 2 x
m

 sin x cos m −1 x (cos dx ), set every cos 2 x = 1 − sin 2 x

n
m odd (+ve) integer
n , m both even (+ve) 1 1
set sin 2 x = (1 − cos 2x ), cos 2 x = (1 − sin 2x )
integer 2 2
2.  tan x sec x dx
n m

 tan
n −1
n odd (+ve) integer x sec m −1 x (sec x tan x dx ), every tan 2 x = sec 2 x − 1

 tan x secm − 2 x (sec2 x dx ), every sec2 x = 1 + tan 2 x

n
m even (+ve) integer
3.  sin ax cos bx dx
1
sin ax cos bx = ( sin (a − b) x + sin (a + b) x )
2
1
cos ax cos bx = ( cos(a − b) x + cos (a + b) x )
2
1
sin ax sin bx = ( cos(a − b) x − cos (a + b) x )
2

Partial fractions  QP ((xx )) dx

f (x ) A B C
 + +
( x − a )( x + b )( x − c ) x − a x +b x −c
f (x ) A Bx + C
 + 2
(x − a )(x 2 + bx + c ) x − a (x + bx + c )
f (x ) A1 A2 An
 + ............. +
(x − a )n x − a (x − a ) 2
(x − a ) n
f (x ) A1x + B 1 A x + B2 A x + Bn
 + 22 ............. + 2 n
( x + bx + c ) n
2
x + bx + c (x + bx + c )
2 2
(x + bx + c ) n

pg. 5
 Trigonometric substitution
f (x ) = a sin 

(a − f  (x ) )
 n

f (x ) = a tan 

(a + f  (x ) )
 n

f (x ) = a sec 

(f (x ) − a  )
 n

 u dv = u v − v du Integration by Parts

L I A T E
‫دوال لوغاريتمية‬ ‫دوال عكسية‬ ‫دوال جبرية‬ ‫دوال مثلثية‬ ‫دوال أسية‬
ln x sin −  x sin x ex
log x cos −  x ch x x
.. .. .. .. ..
. . . .
dv ‫ واالخري تكون ال‬u ‫في االغلب يتم اختيار الدالة التي تسبق في الترتيب ب‬

pg. 6
 Limits
L’hopital rule
f (x )  
y = lim = or
x →a g ( x )  
f (x )
then y = lim
x →a g ( x )

Undefined values
 
, ‫إستخدام أوييتال مباشرة‬
 
. ‫تحول أولا لبسط ومقام ثم أوييتال‬
- ‫يتم توحيد المقام ثم أوييتال‬
‫( للطرفين أولا ثم الوصول ألحد األشكال‬ln) ‫يتم أخذ‬
0 0 ,  0 , 1
‫السابقة‬
Important rules .1
sin x tan x
lim x = 
x →
lim x =  x →

lim f . g = lim f  lim g

x →a x →a x →a
lim ln f (x ) = ln lim f (x )
x →a x →a

lim (+ f (x ) ) = e
g (x ) k

x →a

k = lim f (x ).g (x )
x →a

if f (x ) → , g (x ) → 

pg. 7
 Graphs of Inverse Functions
sin (x ) sin − (x )

 
Domain [ − , ] ‫تحديد مجال‬ Domain [−,]
 
Range [−,]  
Range [ − , ]
 
cos (x ) cos − (x )

Domain [,  ] ‫تحديد مجال‬

Range [−,] Domain [−,]
Range [,  ]

pg. 8
tan (x ) tan − (x )

 
Domain [ − , ] ‫تحديد مجال‬ Domain R
   
Range R Range [ − , ]
 
sec (x ) sec − (x )

 Domain x , x  −
Domain [,  ] − ‫تحديد مجال‬ 
 Range [,  ] −
Range x , x  − 

pg. 9
csc (x ) csc − (x )

Domain x , x  −
 
Domain [− , ] −  ‫تحديد مجال‬  
  Range [− , ]−
 
Range x , x  −
cot (x ) cot − (x )

 
Domain [− , ] −  ‫تحديد مجال‬ Domain R
 
 
Range R Range [− , ]−
 

pg. 10
sh (x ) sh − (x )

Domain R Domain R
Range R Range R
ch (x ) ch − (x )

Domain R + ‫تحديد مجال‬ Domain [, [

Range [, [
Range R +

pg. 11
tanh (x ) tanh − (x )

Domain ] −, [
Domain R
Range ] −, [ Range R

sech (x ) sech − (x )

Domain R + Domain ] ,]

Range ] ,]
Range R +

pg. 12
csch (x ) csch − (x )

Domain R − 
Domain R − 
Range R − 
Range R − 
coth (x ) coth − (x )

-1

Domain R −  Domain x  , x  
Range x  , x   Range R − 

Amr Khaled

pg. 13