Formula Sheet — Maths for Science and Engineering
Table of common integrals

Indices and logarithms b a = c ⇔ loga c = b

´

f (x)

Trigonometry

sin x cos x

sin x + C

2

B

f (x) dx

− cos x + C

sec x cosec x cot x

c

tan x + C
−cosec x + C

sec x tan x

a

sec x + C

2

cosec x
A

− cot x + C

1 x x

C

ln |x| + C ex + C

e

b

ax a b c =
=
(Sine Rule) sin A sin B sin C
2
2
2
a = b + c − 2bc cos A
(Cosine Rule)
Area = 1 ab sin C
2
Trigonometic Identities sin(A ± B)

=

cos(A ± B)

=

tan(A ± B)

=

2

1
(1
2

sin θ =

sin A cos B ± cos A sin B

cos A cos B ∓ sin A sin B tan A ± tan B
1 ∓ tan A tan B
2

− cos 2θ)

cos θ =

1
(1
2

+ cos 2θ)

Arithmetic series a + (a + d) + (a + 2d) + (a + 3d) + . . . nth term

=

Sn

=

a + (n − 1)d

1 n [2a +
2
2
3

(n − 1)d]

Geometric series a + ar + ar + ar + . . . nth term

=

Sn

=

S∞

=

Binomial Series

ar n−1 a(1 − r n )
1−r
a provided that |r| < 1
1−r

n n (a + x)
= a + xan−1 + x2 an−2 + . . . + xn
1
2 n(n − 1) 2 n(n − 1)(n − 2) 3
(1 + x)n = 1 + nx + x + x +...
2!
3!
Maclaurin’s Series f ′′ (0) 2 f ′′′ (0) 3 f ′′′′ (0) 4 x + x + x + ... f (x) = f (0) + xf ′ (0) +
2!
3!
4!
n

1 ax ln a

+C

Integration by parts
ˆ
ˆ dv du u dx = uv − v dx dx dx Trapezium rule
ˆ b f (x) dx ≈ 1 h [y0 + 2(y1 + y2 + . . . + yn−1 ) + yn ]
2
a

Simpson’s rule
ˆ b f (x) dx a 1 h [y0
3

≈

+ 4(y1 + y3 + . . . + yn−1 )

+2(y2 + y4 + . . . + yn−2 ) + yn ] b−a and n is the number of strips.
Where h = n Newton-Raphson method
To solve f (x) = 0 with an