here

UCC Mathematical Tables
Calculus
f (x)
xn
ln |x|
cos x
sin x
tan x
sec x
cosec x
cot x
ex
eax
ax
x
cos−1
a
−1 x
sin
a
x
tan−1
a
x
sec−1
a
−1 x
cosec
a
Product rule
Quotient rule
Chain rule
Integration by parts
Trapezoidal rule
f ′ (x)
nxn−1
1
x
− sin x
cos x
sec2 x
sec x tan x
− cosec x cot x
− cosec2 x
ex
aeax
ax ln a
1
−√
a2 − x2
1
√
2
a − x2
a
a2 + x2
a
√
x x2 − a2
a
− √
2
x x − a2
f (x)
−1
cot
x
a
sinh x
cosh x
tanh x
sech x
cosech x
coth x
sinh−1 x
cosh−1 x
tanh−1 x
sech−1 x
cosech−1 x
coth−1 x
f ′ (x)
f (x)
a
− 2
a + x2
cosh x
sinh x
sech2 x
− sech x tanh x
− cosech x coth x
− cosech2 x
1
√
2
x +1
1
√
x2 − 1
1
1 − x2
1
− √
x 1 − x2
1
− √
x x2 + 1
1
− 2
x −1
dy
dv
du
=u
+v
dx
dx
dx
dv
du
v
−u
u
dy
dx
dx
y=
⇒
=
v
dx
v2
du dv
f (x) = u v(x) ⇒ f ′ (x) =
dv dx
Z
Z
u(x)v ′ (x) dx = uv − v(x)u′ (x) dx
y = uv ⇒
y1
Taylor series (centre a)
Maclaurin series
Volume of solid of
revolution about x-axis
Simpson’s rule (n odd)
y2
h
y3
f (x) dx
xn+1
xn (n 6= −1)
n+1
1
ln |x|
x
cos x
sin x
sin x
− cos x
tan x
ln | sec x|
sec x
ln | sec x +
tan x|
x cosec x
lntan 2
cot x
ln | sin x|
ex
ex
1 ax
eax
e
ax
a
ax
ln a
x
1
√
sin−1
2
2
a
a −x
1
1
−1 x
tan
x2 + a2
a
a
1
1
x
−1
√
sec
a
a
x x2 − a2
Newton-Raphson
i
hh
A≈
y1 + yn + 2(y2 + y3 + · · · + yn−1 )
2
R
y4
xn+1 = xn −
f (x)
1
√
x2 + a2
1
2
a − x2
1
√
x2 − a2
sinh x
cosh x
tanh x
coth x
sech x
cosech x
cos2 x
sin2 x
cosh2 x
sinh2 x
1
√
2
x a − x2
1
√
2
x x + a2
R
f (x) dx
x + √x2 + a2 ln
a
1 a + x
ln
2a
a−x
x + √x2 − a2 ln
a
cosh x
sinh x
ln cosh x
ln | sinh x|
−1
tan
x)
(sinh
x lntanh 2
1
1
2 (x + 2 sin 2x)
1
1
2 (x − 2 sin 2x)
1
1
2 (x + 2 sinh 2x)
1
1
2 (−x + 2 sinh 2x)
1
x
− sech−1
a
a
1
−1 x
− cosech
a
a
f (xn )
f ′ (xn )
f ′′ (a) 2
f (r) (a) r
x + ···+
x + ···
2!
r!
f ′′ (0) 2
f (r) (0) r
f (x) = f (0) + f ′ (0)x +
x + ···+
x + ···
2!
r!
Z x=b
V =
πy 2 dx
f (a + x) = f (a) + f ′ (a)x +
x=a
i
hh
A≈
y1 + yn + 2(y3 + y5 + · · · + yn−2 ) + 4(y2 + y4 + · · · + yn−1 )
3
yn
Trigonometry
sin A
cos A
cos A
1
cot A =
=
sin A
tan A
1
sec A =
cos A
1
cosec A =
sin A
tan A =
cos2 A + sin2 A = 1
cos 2A = cos2 A − sin2 A
2 cos A cos B = cos(A + B) + cos(A − B)
sec2 A = 1 + tan2 A
sin 2A = 2 sin A cos A
2 sin A cos B = sin(A + B) + sin(A − B)
cos(A + B) = cos A cos B − sin A sin B
sin(A + B) = sin A cos B + cos A sin B
tan A + tan B
1 − tan A tan B
cos(−A) = cos A
tan(A + B) =
sin(−A) = − sin A
cos(A − B) = cos A cos B + sin A sin B
tan(−A) = − tan A
sin(A − B) = sin A cos B − cos A sin B
einθ = (cos θ + i sin θ)n
tan(A − B) =
= cos nθ + i sin nθ
1
(1 + cos 2A)
2
1
sin2 A = (1 − cos 2A)
2
2 tan A
tan 2A =
1 − tan2 A
1 − tan2 A
cos 2A =
1 + tan2 A
2 tan A
sin 2A =
1 + tan2 A
cos2 A =
2 sin A sin B = cos(A − B) − cos(A + B)
2 cos A sin B = sin(A + B) − sin(A − B)
A+B
A−B
cos
2
2
A+B
A−B
cos A − cos B = −2 sin
sin
2
2
A+B
A−B
sin A + sin B = 2 sin
cos
2
2
A+B
A−B
sin A − sin B = 2 cos
sin
2
2
cos A + cos B = 2 cos
tan A − tan B
1 + tan A tan B
Length/area/volume
Triangle
A
c
b
h
C
B
a
p
Area A = 12 ab sin C = 12 ah = s(s − a)(s − b)(s − c),
where s = 12 (a + b + c)
a
b
c
Sine rule:
=
=
sin A
sin B
sin C
Cosine rule: a2 = b2 + c2 − 2bc cos A
Right-angled triangle
a
c
b
cos A =
c
a
tan A =
b
c2 = a 2 + b 2
sin A =
c
a
A
b
b
Parallelogram
b
h
Area A = ah = ab sin C
Trapezium
h
Area A =
a + b
h
2
C
a
a
l
Circle
r
Circumference l = 2πr
Area A = πr2
Arc/sector
Curved surface area A = 2πrh
Volume V = πr2 h
Cone
θ
Length l = rθ
Area A = 12 r2 θ
r
l
(θ in radians)
r
Cylinder
h
l
h
Curved surface area A = πrl
Volume V = 13 πr2 h
r
r
Sphere
r
Surface area A = 4πr
Volume V = 43 πr3
2
Frustrum of cone
l
h
R
Curved surface area A = π(r + R)l
Volume V = 13 πh(R2 + Rr + r2 )