Revision Notes For Core Page 2
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1
2
3
4Rational Functions and Partial Fractions
a
c
To do
x
factorise anything that you can then cancel down.
b
d
a
c
To do
÷
turn second fraction upside down then multiply.
b
d
ad +
a
c
bc
To do
+
make one fraction by cross multiplying so
or make a common
b
d
bd
denominator then combine.
To do the reverse (make two separate fractions from one), factorise the denominator
into two factors, express as two fractions each with one of the factors as the
denominator and A and B as the numerators, cross multiply then solve for A and B.
−
−
−
+
−
7
x
8
A
B
7
x
8
A
x (
) 2
B
2 (
x
) 1
e.g.
=
+
then
=
then set
−
−
−
−
−
−
−
−
2 (
x
1
)(
x
) 2
2
x
1
x
2
2 (
x
1
)(
x
) 2
2 (
x
1
)(
x
) 2
1
x=
to find A and x=2 to find B.
2
2
If there is a repeated root (e.g. denominator may be (2x+3)(x-2)
then 3 fractions
A
B
C
+
+
+
−
2 )
2
x
3
x
2
−
x (
2
Multiply top and bottom by 2x + 3 then A is on its own and B and C have (2x+3) on the
top then make x= - 3/2 and this will eliminate B and C to find A. Now do the same for (x
– 2) to find B and so on.
Division of Polynomials
Use long division to find the quotient (how many times it goes in) and the remainder then
can rearrange and thus integrate fractions with polynomials in them.
3
2
(20x
- x
– 4x – 7) ÷ (4x + 3) =
2
−
+
5
x
4
x
2
(quotient)
3
2
−
−
−
4x+3
20x
x
4
x
7
3
20x
+ 15x
2
-16x
-4x-7
2
-16x
-12x
8x - 7
8x+6
-13 (remainder)
∫
∫
3
2
2
So
(20x
- x
– 4x – 7) dx=
(5x
- 4x + 2) – 13 dx and can integrate the right hand
(4x + 3)
(4x + 3) side so find the solution to the
left hand side.
Differential Equations
Get the x’s and dx together on one side and the y’s and dy on the other then integrate
either side. This gives you an equation in terms of x and y rather than dx and dy. Then
use the information you are given to find the constant of integration then can use the
equation to solve the problem. Put the constant of integration on one side only.
dy
1
2
Example
If
=y
x then
dy =
x dx
2
dx
y
−
3
1
1
2
∫
∫
dy =
x dx therefore
=
x
2
+ c
2
y
3
y
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