Download Formulas and Methods for A2 Differentiation and Integration

A2 Differentiation and Integration Methods.
Differentiation
Function
yx
n
y  ax n
y  ax n  bx m
y  (ax n  b)m
Differential
dy
dx
dy
dx
dy
dx
dy
dx
 nx n 1
 anx n 1
 anx n 1  bmx m 1
 manx n 1 (ax n  b) m 1
y  ( f ( x))n
dy
 nf '( x)( f ( x)) n 1
dx
y  ln x
dy 1

dx x
dy f '( x)

dx f ( x)
dy
 ex
dx
y  ln f ( x)
y  ex
y  e f ( x)
y  ax
y  sin x
y  cos x
dy
dx
dy
dx
dy
dx
dy
dx
 f '( x)e f ( x )
Example
dy
y  x5 
 5x4
dx
dy
y  3x 4 
 12 x3
dx
dy
y  2 x 4  5 x3 
 8 x 3  15 x 2
dx
dy
y  (2 x3  4)5 
 30 x 2 (2 x3  4) 4
dx
y  (2 x5  3x7 )3  3(10 x 4  21x6 )(2 x5  3x7 )2
y  ln 5 x 2 
y  e2 x 
3
dy 10 x 2


dx 5 x 2 x
3
dy
 6 x 2e2 x
dx
 a x ln a
 cos x
  sin x
1
Notes
If you differentiate a constant you get zero
You can differentiate polynomials term by
term.
Multiply by the power of the bracket.
Multiply by the differential of the
function in the bracket.
Drop the power of the bracket by 1.
This is the general formula that works for
polynomials and later functions. (E.g. see
sinnf(x))
Function differentiated divided by
function.
A special function whose gradient at any
point is equal to the value of the function
at that point.
e to the function of x multiplied by the
differential of the function.
When a=e, lne=1 so result reduces to that
above for ex.
A2 Differentiation and Integration Methods.
y  tan x
y  sin nx
y  cos nx
y  tan nx
y  sin f ( x)
y  cos f ( x)
y  tan f ( x)
y  sec x
y  cot x
y  cos ecx
y  sin n f ( x)
y  cos n f ( x)
y  tan n f ( x)
y  sin 1 x
dy
dx
dy
dx
dy
dx
dy
dx
dy
dx
dy
dx
dy
dx
dy
dx
dy
dx
dy
dx
dy
dx
 sec 2 x
 n cos nx
  n sin nx
 n sec 2 nx
 f '( x) cos f ( x)
  f '( x) sin f ( x)
 f '( x) sec 2 f ( x)
dy
 3cos 3 x
dx
dy
y  cos 5 x 
 5sin 5 x
dx
dy
y  tan 2 x 
 2sec 2 2 x
dx
dy
y  sin 2 x 7 
 14 x 6 cos 2 x 7
dx
dy
y  cos 3x 6 
 18 x5 sin 3 x 6
dx
dy
y  tan 3x 4 
 12 x3 sec 2 3x 4
dx
y  sin 3x 
Differential of function multiplied by cos
of function.
Differential of function multiplied by
differential of sin of function.
Differential of function multiplied by
differential of tan of function.
 sec x tan x
  cos ec 2 x
  cos ecx cot x
 nf '( x) cos f ( x) sin n 1 f ( x)
y  sin 5 3x 2 
dy
 30 x cos 3x 2 sin 3x 2
dx
dy
 nf '( x) sin f ( x) cos n 1 f ( x)
dx
dy
 nf '( x) sec 2 f ( x) tan n 1 f ( x)
dx
dy
1

dx
1  x2
2
Multiply by the power.
Multiply by the differential of the
function
Multiply by the differential if sin.
Drop the power by one.
A2 Differentiation and Integration Methods.
dy
1

dx
1  x2
dy
1

dx 1  x 2
dy
dv
du
u v
dx
dx
dx
du
dv
v
u
dy
 dx 2 dx
dx
v
y  cos1 x
y  tan 1 x
y  uv
y
u
v
dy
  x cos x  sin x
dx
x
dy sin x  x cos x
y


sin x
dx
sin 2 x
y  x sin x 
Product rule
Quotient rule.
Integration
Function
Integral
 x dx
x
c
n 1
 ax dx
ax n 1
c
n 1
ax n 1 bx m1

c
n 1 m 1
(ax n  b)m  c
n
 ax
n
 bx m dx
 manx
n 1
(ax n  b)m1dx
 nf '( x)( f ( x))
1
 x dx
Example
n 1
n
n 1
dx
( f ( x))n  c
 x dx 
3
4
x
c
4
2 x5 5 x 4
 2 x  5x dx  5  4  c
2
3
4
3
5
 30 x (2 x  4) dx  (2 x  4)  c
4
 3(10 x
3
4
 21x 6 )(2 x5  3x 7 ) 2 dx  (2 x5  3x 7 )3  c
ln x  c
3
Notes
Raise the power by 1 and divide by
the new power.
Does not work when n=-1, this give
the special case lnx.
You can integrate polynomials term
by term.
Realise that the differential of the
bracket is outside the bracket.
This is the general formula.
Realise that the differential of the
bracket is outside the bracket.
x0
Consider  x 1dx   c using the
0
basic rule for integration. This
would imply that the area is always
infinite beneath a 1/x graph, which
is clearly ridiculous. Hence the
A2 Differentiation and Integration Methods.

ln f ( x)  c
f '( x)
dx
f ( x)
 e dx
ex  c
e
eax
a
ax
c
ln a
x ln x  x  c
x
ax
dx
 a dx
x
 ln xdx
 sin xdx
 cos xdx
 tan xdx
 sin nxdx
10 x
2
 5x 2 dx  ln 5 x
3 x
 e dx  
e3 x
c
3
special case.
Spot, function differentiated divided
by function.
Since the differential of ex is ex,
then the integral of ex is ex (+c).
e to the ax divided by a..
If a=e this reduces to the ex result
above.
 cos x  c
sin x  c
 ln sec x  c
 cos nxdx
 sec xdx
 cot xdx
 cos ecxdx
ln can only take positive values.
 cos nx
c
n
 sin 3xdx 
 cos 3 x
c
3
sin nx
c
n
ln s ec x  tan x  c
 cos 5xdx 
sin 5 x
c
5
Integrating sinnx and cosnx is
relatively easy compared to
integrating powers of sin and cos.
ln sin x  c
 nf '( x) cos f ( x) sin
n 1
 nf '( x) sin f ( x) cos
 nf '( x) sec f ( x) tan
n 1
2
f ( x)dx
dy
 ln cos ecx  cot x  c
dx
sin n f ( x)  c
f ( x)dx
cosn f ( x)  c
n 1
f ( x)dx
30 x cos 3x 2 sin 3x 2  sin 5 3x 2  c
tan n f ( x)  c
4
Recognise that you have a function
multiplied by its differential.
A2 Differentiation and Integration Methods.


1
1  x2
1
sin 1 x  c
dx
cos 1 x  c
dx
1  x2
1
 1  x 2 dx
dv
 u dx dx
Notice similarity between sin and
cos forms.
tan 1 x  c
uv   v
Integration by parts.
du
dx
dx
Products of Trig Functions
Integral
sinn x cosm x dx, n is odd
Method
Factorise one sine out and convert the remaining
sines to cosines using sin2 x 1cos2 x.
Example
 sin x cos xdx
  sin x(1  cos x) cos
  sin x  sin x cos x
3
2
2
4
1
  cos x  cos5 x  c
5
sinn x cosm x dx, m is odd
sinn x cosm x dx n and m are both odd
sinn x cosm x dx n and m are both even.
tann x secm x dx, n is odd.
Factorise one cosine out and convert the
remaining cosines to sines
using cos2 x 1sin2 x.
Use either 1. or 2.
Use double angle formula for sine and/or half
angle
formulas to reduce the integral into a form that
can be integrated.
Factorise one tangent and one secant out and
convert the remaining tangents to secants using
tan2 xsec2 x 1.
5
2
x
A2 Differentiation and Integration Methods.
tann x secm x dx, m is even.
Factorise two secants out and convert the
remaining secants to tangents using sec2 x
1tan2 x .
 tan x sec xdx
  sec x(1  tan x) tan
  sec x  sec tan x
3
4
2
2
2
2
5
1
 tan x  tan 6 x  c
6
tann x secm x dx, n is odd and m is even.
Use either 1. or 2.
tann x secm x dx, n is even and m is odd.
Each integral will be dealt with differently
6
3
x