DIFFERENTIATION FORMULAS
BASIC ALGEBRAIC DIFFERENTIATION
d n
du
▪
u = nu n −1
dx
dx
d
(Cu ) = C du
▪
dx
dx
d
du dv
u±v=
±
▪
dx
dx dx
d
dv
du
uv = u
+v
▪
dx
dx
dx
du
dv
v
−u
d u
dx
dx
▪
 =
dx  v 
v2
d  C  −C dv
▪
⋅
 =
dx  v  v2 dx
d  v  1 dv
▪
 = ⋅
dx  C  C dx
du
d
▪
u = dx
dx
2 u
1
 1
d  n  1 n −1 du
▪
u
=
⋅
u
⋅
dx   n
dx
 
LOGARITHMIC DIFFERENTIATION
d
(log a u ) = 1 log a e du
▪
dx
u
dx
1 du
d
▪
(ln u ) = ⋅
dx
u dx
EXPONENTIAL DIFFERENTIATION
( )
▪
( )
d u
du
e = eu
dx
dx
BASIC TRIGONOMETRIC INTEGRATION
▪
▪
▪
▪
▪
∫
∫ cos udu = sin u + C
2
∫ sec udu = tan u + C
2
∫ csc udu = − cot u + C
▪
( )
d u
du
C = C u ⋅ lnC ⋅
dx
dx
TRIGONOMETRIC DIFFERENTIATION
d
▪
(sin u ) = cos u ⋅ du
dx
dx
d
(cos u ) = − sin u ⋅ du
▪
dx
dx
d
du
2
(tan u ) = sec u ⋅
▪
dx
dx
d
2
(cot u ) = − csc u ⋅ du
▪
dx
dx
d
(sec u ) = sec u tan u ⋅ du
▪
dx
dx
d
(csc u ) = − csc u cot ⋅ du
▪
dx
dx
INVERSE TRIGONOMETRIC DIFFERENTIATION
d
du
1
▪
⋅
sin −1 u =
2 dx
dx
1− u
d
−1
du
▪
cos−1 u =
⋅
2
dx
1 − u dx
(
(
)
(
(
(
)
)
d
1
du
tan −1 u =
⋅
dx
1 + u 2 dx
d
− 1 du
▪
cot −1 u =
⋅
dx
1 + u 2 dx
d
1
du
▪
sec−1 u =
⋅
2
dx
u u − 1 dx
▪
sin udu = − cos u + C
∫
)
)
IF BOTH ARE ODD,
TRANSFORM LOWER POWER
(
INVERSE HYPERBOLIC DIFFERENTIATION
d
1
du
▪
sinh−1 u =
⋅
2
dx
u + 1 dx
sin n u = (sin 2 u )
n2
CASE 2:
n2
2
(
)
(
(
(
)
)
)
∫ secn =uduEVENor ∫ csc udu
n
n
n−2
n
2
sec u = sec
u (sec u )
2
1
2
Use sin u =
(1 − cos2u ) or
CASE 3: tan m u sec n udu or
2
csc u cot udu = − csc u + C
▪
1
cos2 u = (1 + cos 2u )
cot m u csc n udu
▪
tan udu = − ln cos u + C
2
n IS EVEN;
CASE 4: sin mu cos n udu
SIMILAR TO CASE 2
▪
tan udu = ln sec u + C
m & n ARE EVEN, TRANSFORM BOTH
CASE 4: tan m u sec n udu or
SAME AS CASE 3
▪
cot udu = ln sin u + C
CASE 5:
cot m u cscn udu
▪
sec udu = ln sec u + tan u + C ▪ sinα cosβ = 1 [sin(α + β ) + sin(α − β )]
m = ODD
2
m
n
tan u sec udu =
▪
csc udu = ln csc u − cot u + C ▪ cosα cosβ = 1 [cos(α + β) + cos(α − β)]
m−1
2
tan
u secn−1 u(tanu secudu)
TRANSFORMATIONS OF TRIGONOMETRIC
1
transform
▪ sinα sinβ = − [cos(α + β) − cos(α − β)]
FUNCTIONS
2
cotm u cscn udu =
SINE/COSINE
CASE 1: n = ODD
m −1
TANGENT/COTANGENT/SECANT/COSECANT
cot
u cscn −1 u(cot u cscudu)
n
n −1
n
n
CASE
1:
or
tan udu
cot udu
transform
sin u = sin
u (sin u )
∫
∫
∫
∫
∫
∫
cos u
n
n −1
cos u = cos
u (cos u )
tan u = tan
n−2
u (tan 2 u )
transform
Use sin 2 u + cos 2 u = 1
CASE 2: sin m u cos n udu
ONE EXPONENT IS ODD;
SIMILAR TO CASE 1
∫
cot n u = cot n − 2 u (cot 2 u )
2
2
transform
where: tan u = sec u −1
cot2 u = csc2 u − 1
∫
∫
∫
x n +1
∫ x dx = n + 1 + C ; n ≠ -1,n = constant
n
INTEGRANDS YIELDING TO NATURAL LOGARITHMS
du
▪
= ln u + C
u
EXPONENTIAL INTEGRATION
∫
au
+C
▪ eu du = eu + C ▪ au du =
ln a
where: e = 0.718 = constant
a = constant other than e
DEFINITE INTEGRALS
∫
∫
b
▪
∫a f (x)dx = f (x)]a = f (b) − f (a)
b
b
▪
a
∫a f (x )dx = −∫b f (x)dx
∫ f (x)dx = ∫ f (x)dx + ∫ f (x)dx
b
c
b
a
a
c
b
b
a
a
∫ f (x )dx = ∫ f (y )dy
du
∫u
2
u −a
=
2
where: a = lower limit
b = upper limit
1
u
sec −1 + C
a
a
BASIC HYPERBOLIC INTEGRATION
▪
▪
∫ sinh udu = cosh u + C
∫ cosh udu = sinh u + C
2
∫ sec h udu = tanh u + C
▪
∫ csch udu = −cothu + C
▪
∫ sec hu tanh udu = − sec hu + C
▪
∫ csc hu coth udu = − csc hu + C
2
INTEGRALS YIELDING TO INVERSE HYPERBOLIC FUNCTIONS
du
u
▪
= sinh −1 + C
2
2
a
u +a
du
−1 u
= cosh
+C
▪
a
u 2 − a2
∫
∫
▪
▪
INTEGRALS LEADING TO INVERSE TRIGONOMETRIC ▪
FUNCTIONS
du
u
▪
▪
= sin−1 + C
a
a2 − u 2
du
1
u
▪
= tan−1 + C
a
a2 + u 2 a
∫
∫
GENERAL POWER FORMULA
∫
n
sin u
∫
∫
2
where: sec u = 1 + tan u
csc2 u = 1 + cot2 u
transform
∫
∫
∫
∫
∫
∫
▪ ∫ af (x )dx = a ∫ f ( x )dx
; a = constant
1
▪
f (x )dx =
af (x )dx
a
▪ [ f (x ) ± g (x )]dx = f (x )dx ± g (x )d x
2
csc u = csc
u (csc u )
)
∫ dx = x + C
▪
▪
n−2
n
(
d
−1
du
csc h −1u =
⋅
2
dx
u 1 + u dx
INTEGRATION FORMULAS
BASIC INTEGRATION
▪
transform
cos u = (cos u)
sec u tan udu = sec u + C
)
transform
transform
n
(
d
1
du
cosh−1 u =
⋅
2
dx
u − 1 dx
d
1
du
▪
tanh−1 u =
⋅
;1 > u2
dx
1 − u 2 dx
d
−1 du
▪
coth−1 u = 2 ⋅ ; u 2 > 1
dx
u − 1 dx
d
−1
du
−1
▪
sec h u =
⋅
2 dx
dx
u 1− u
▪
CASE 3: n = EVEN
)
d
−1
du
csc −1 u =
⋅
2
dx
u u − 1 dx
HYPERBOLIC DIFFERENTIATION
d
▪
(sinh u ) = cosh u ⋅ du
dx
dx
d
du
▪
(cosh u ) = sinh u ⋅
dx
dx
d
2
▪
(tanh u ) = sec h u ⋅ du
dx
dx
d
du
2
▪
(coth u ) = − csch u ⋅
dx
dx
d
▪
(sec hu ) = − sec hu tanh u ⋅ du
dx
dx
d
du
(csc hu ) = − csc hu coth u ⋅
▪
dx
dx
▪
du
−1 u
∫ a2 − u 2 = tanh a + C ,|u|<a
du
1
u
∫ u − a = − a coth a + C , u >a
−1
2
∫u
∫u
2
du
2
a −u
2
du
2
a +u
2
=
−1
u
sec h −1 + C
a
a
=
−1
u
csc h−1 + C
a
a
2
2
WALLI’S FORMULA
π 2
∫0
sinm u cosn u =
[(m − 1)(m − 3)(m − 5)...m = 2or1][(n − 1)(n − 3)(n − 5)...n = 2or1]
•α
(m + n)(m + n − 2)(m + n − 4)...m + n = 2or1
where: α = π ; if both m & n are even
2
α = 1 ; if otherwise
TECHNIQUES OF INTEGRATION
INTEGRATION BY PARTS
▪ sin u =
∫ udv = uv − ∫ vdu
▪ cos u =
INTEGRATION BY TRIGONOMETRIC
SUBSTITUTION
CASE 1: a 2 − u 2
▪ u = a sinθ
▪ du = a cosθdθ
CASE 2: a 2 + u 2
▪ u = a tanθ
▪ du = a sec2 θdθ
CASE 3: u 2 − a 2
▪ u = a secθ
▪ du = a secθ tanθdθ
▪ dx =
▪
▪
z
1
du
u−a
∫ u 2 − a 2 = 2a ln u + a
du
∫ a2 − u
=
2
+C
1
a+u
ln
+C
2a a − u
∫a
A = LW ; L = length, W = width
b
▪ A=
∫ ( yh − yl )dx
b
▪ A=
2
∫a (xr − xl )dy ; x = x on the right
r
xl = x on the left
log a a = 1
log a ( xy ) = log a x + log a y
x
log a   = log a x − log a y
 y
log a y x = x log a y
1
log a y
x
log a a x = x
y=
log a x
a
=x
EXPONENTIAL FUNCTIONS
V =π
b
∫a r h ; r = radius, h = height
2
e
ln x
=x
HYPERBOLIC FUNCTIONS
e x − e− x
2
e x + e− x
cosh x =
2
sinh x =
∫a
V
b
Mxz
▪ y=
; Mxz = ∫ yc dV
a
V
▪ z=0
▪ S=
cosh 2 x = cosh 2 x + sinh 2 x
INVERSE HYPERBOLIC FUNCTIONS EXPRESSED AS
LOGARITHMIC FUNCTIONS
(
x = ln ( x ±
sinh −1 x = ln x + x 2 + 1
cosh
−1
xc dV
 dx 
 dy ; x = g(x)
 dy 
1+ 
a
b x2
▪
∫∫
▪
∫a ∫y
1st f (x , y )dx dy
a x1
2 nd
assume y = constant
1st b y2 1
f (x , y )dy dx
assume x = constant
2 nd
AREAS BY DOUBLE INTEGRATION
b x2
▪
∫a ∫x dxdy
b y
∫a ∫y dydx
1
2
▪
1
note: the centroid always lies on the axis of rotation
LENGTH OF AN ARC – S
cosh x − sinh x = e− x
sinh 2 x = 2sinh x cosh x
)
x −1)
b
∫
2
b
MULTIPLE INTEGRALS
DOUBLE INTEGRALS
; t = thickness
▪ x = Myz ; Myz =
cosh x + sinh x = e x
tanh 2 x + sec h2 x = 1
sinh( x + y ) = sinh x cosh y + cosh x sinh y
cosh( x + y ) = cosh x cosh y + sinh x sinh y
a x = e x ln a
ri = inner radius
b
▪ x = My ; My = xc dA
∫
a
A
b
▪ y = Mx ; Mx = y dA
∫a c
A
note: the centroid lies on axis of symmetry
cosh 2 x − sinh 2 x = 1
a x+ y = a x ⋅ a y
a log a x = x
− ri 2 ) h ; ro = outer radius
CENTROID OF AREAS C ( x , y )
e x − e− x
e x + e− x
e x + e− x
coth x = x − x
e −e
2
sec hx = x − x
e +e
2
csc hx = x − x
e −e
BASIC HYPERBOLIC FORMULAS
coth 2 x = csc h2 x + 1
ax
ay
(a x )r = a rx
b
∫arht
V = 2π
tanh x =
a0 = 1
a x− y =
2
CENTROID OF VOLUMES C ( x , y , z )
; yh = upper y
yl = lower y
a
2
b
∫a ( ro
▪ S=
VOLUME SHELL METHOD
VOLUME DISK METHOD
2 dz
log a 1 = 0
x
2
b
−dz
PROPERTIES OF LOGARITHM
If a y = x then log a x = y ;
log a
V =π
AREA BETWEEN TWO CURVES
1
▪ z = tan x
2
1+ z
1+ z
VOLUME RING METHOD
APPLICATIONS
AREA UNDER A CURVE
INTEGRATION OF RATIONAL FUNCTIONS OF
SIN & COS
▪ du =
1+ z2
1 − z2
OTHER INTEGRATION FORMULAS
RECIPROCAL SUBSTITUTION
1
▪ x=
z
2z
2
1  1 + x  sec h −1 x = ln  1 ± 1 − 1 
tanh −1 x = ln 
x

x 
2  1− x 

1

1
1  x +1 
−1
coth −1 x = ln 
 csc h x = ln  x ± x 2 + 1 
2  x −1 


∫
b
a
2
 dy 
 dx ; y = f(x)
 dx 
1+ 
TRIGONOMETRIC IDENTITIES
▪ RECIPROCAL IDENTITIES
1
sin θ
1
sec θ =
cos θ
1
cot θ =
tan θ
tan( x − y ) =
tan x − tan y
1 + tan x tan y
1 + tan 2 θ = sec 2 θ
DOUBLE ANGLE FORMULAS
sin 2θ = 2 sin θ cos θ
cos 2θ = cos 2 θ − sin 2 θ
cos 2θ = 1 − 2 sin 2 θ
2 tan θ
tan 2θ =
1 − tan 2 θ
POWER REDUCING FORMULAS
1 − cos 2θ
sin 2 θ =
2
1 + cos 2θ
2
cos θ =
2
1 − cos 2θ
2
tan θ =
1 + cos 2θ
HALF-ANGLE FORMULAS
1 + cot 2 θ = csc 2 θ
sin
csc θ =
TANGENT AND COTANGENT
sin θ
tan θ =
cos θ
cos θ
cot θ =
sin θ
PYTHAGOREAN IDENTITIES
sin 2 θ + cos 2 θ = 1
COFUNCTION IDENTITIES
sin(90 − θ ) = cos θ
cos
csc ( 90 − θ ) = sec θ
sec ( 90 − θ ) = csc θ
tan
cos ( 90 − θ ) = sin θ
tan
cot ( 90 − θ ) = tan θ
SUM & DIFFERENCE FORMULAS
sin( x + y ) = sin x cos y + cos x sin y
sin( x − y ) = sin x cos y − cos x sin y
cos( x + y ) = cos x cos y − sin x sin y
cos( x − y ) = cos x sin y + sin x sin y
tan x + tan y
tan( x + y ) =
1 − tan x tan y
θ
cot
cot
2
θ
2
θ
2
θ
2
θ
2
θ
2
=
1 − cos θ
2
1 + cos θ
2
1 − cos θ
=
sin θ
sin θ
=
1 + cos θ
sin θ
=
1 − cos θ
1 + cos θ
=
sin θ
=
SINE LAW
a
b
c
=
=
sin A sin B sin C
COSINE LAW
a 2 = b 2 + c 2 − 2bc cos A
b 2 = a 2 + c 2 − 2ac cos B
c 2 = a 2 + b 2 − 2ab cos C
LAW OF TANGENTS
1
tan ( A − B)
a −b
2
=
a + b tan 1 ( A + B )
2
PRODUCT TO SUM FORMULAS
1
[cos( x − y ) − cos( x + y )]
2
1
cos x sin y = [ cos( x + y ) + cos( x − y ) ]
2
1
sin x cos y = [sin( x + y) + sin( x − y ) ]
2
1
cos x sin y = [sin( x + y) − sin( x − y ) ]
2
sin 2 x
sin x cos x =
2
sin x sin y =
SUM TO PRODUCT FORMULAS
 x+ y
 x− y
sin x + sin y = 2 sin 
 cos 

 2 
 2 
x
+
y
x
−
y

 

sin x − sin y = 2 cos 
 sin 

 2   2 
 x+ y
 x− y
cos x + cos y = 2 cos 
 cos 

 2 
 2 
 x+ y  x− y
cos x − cos y = −2sin 
 sin 

 2   2 
MOLLWEICLE’S EQUATION
1
sin ( A − B)
a−b
2
=
1
C
cos C
2
1
cos ( A − B)
a+b
2
=
1
C
sin C
2