MF 15

List MF15

MATHEMATICS (8864)
MATHEMATICS (9740)
MATHEMATICS (9810)

LIST OF FORMULAE
AND
STATISTICAL TABLES

PURE MATHEMATICS
Algebraic series
Binomial expansion:
n
n
n
(a + b) n = a n +   a n −1b +   a n − 2b 2 +   a n − 3b3 + K + b n , where n is a positive
1
 2
 3
 
 
 
n
n! integer and   =
 r  r!(n − r )!
 

Maclaurin’s expansion:

x2 x n (n) f ′′(0) + K + f (0) + K n! 2! n(n − 1) 2 n(n − 1) K (n − r + 1) r x +K
(1 + x) n = 1 + nx + x +K+
2!
r! f( x) = f(0) + x f ′(0) +

(

x < 1)

x2 x3 xr +
+K+
+K r! 2!
3!

(all x)

sin x = x −

(−1) r x 2 r +1 x3 x5
+
−K+
+K
3!
5!
(2r + 1)!

(all x)

cos x = 1 −

(−1) r x 2 r x2 x4
+
−K+
+K
2!
4!
(2r )!

(all x)

ex =1+ x +

ln(1 + x) = x −

(−1) r +1 x r x2 x3
+
−K+
+K
r
2
3

Partial fractions decomposition
Non-repeated linear factors: px + q
A
B
=
+
(ax + b)(cx + d ) (ax + b) (cx + d )

Repeated linear factors: px 2 + qx + r
(ax + b)(cx + d )

2

=

A
B
C
+
+
(ax + b) (cx + d ) (cx + d ) 2

Non-repeated quadratic factor: px 2 + qx + r
2

2

(ax + b)( x + c )

2

=

A
Bx + C
+ 2
(ax + b) ( x + c 2 )

( − 1 < x Y 1)

Trigonometry sin( A ± B) ≡ sin A cos B ± cos A sin B cos( A ± B) ≡ cos A cos B m sin A sin B

tan( A ± B) ≡

tan A ± tan B
1 m tan A tan B

sin 2 A ≡ 2 sin A cos A cos 2 A ≡ cos 2 A − sin 2 A ≡ 2 cos 2 A − 1 ≡ 1 − 2 sin 2 A

tan 2 A ≡

2 tan A
1 − tan 2 A

sin P + sin Q ≡ 2 sin 1 ( P + Q) cos 1 ( P − Q)
2
2 sin P − sin Q ≡ 2 cos 1 ( P + Q) sin 1 ( P − Q)
2
2 cos P + cos Q ≡ 2 cos 1 ( P + Q) cos 1 ( P − Q)
2
2 cos P − cos Q ≡ −2 sin 1 ( P + Q) sin 1 ( P − Q)
2
2
Principal values:

− 1 π Y=sin−1x Y= 1 π
2
2

( x Y 1)

0 Y cos−1x Y π

( x Y 1)

− 1 π < tan −1 x < 1 π
2
2

Derivatives f ′( x)

f(x)

1

sin −1 x

1− x 2

cos −1 x

−

1
1− x 2

tan −1 x

1
1 + x2

sec x