Mathematical Tables
Trigonometric
sin
tan = cos
cot = tan1
sec = cos1
cosec = sin1
sin( ) = sin
cos( ) = cos
sin(A B) = sin A cos B cos A sin B
cos(A B) = cos A cos B sin A sin B
A tan B
tan(A B) = 1tantan
A tan B
cos A 2 B
cos A + cos B = 2 cos A+B
2
A+B
cos A cos B = 2 sin 2 sin A 2 B
cos
= sin
2
= cos
sin
2
sin (90o
) = cos
cos (90o
) = sin
1
2
cos = 2 [1 + cos (2 )]
sin2 = 21 [1 cos (2 )]
cos3 = 41 [3 cos + cos (3 )]
sin3 = 41 [3 sin
sin (3 )]
sin2 + cos2 = 1
1 + cot2 = cosec2
cos2
sin2 = cos (2 )
2 sin cos = sin (2
p)
a cos + b sin = (a2 + b2 ) cos( + ); where = tan
p
a cos +b sin =
(a2 + b2 ) cos( + ); where = tan
sin A sin B = 21 [cos (A B) cos (A + B)]
cos A cos B = 21 [cos (A B) + cos (A + B)]
sin A cos B = 12 [sin (A B) + sin (A + B)]
Indicies
x0 = 1
xp xq = xp+q
xp
= xp q
xq
(xp )q = xpq
Logs
ln x loge x
log (xp ) = p log x
10 (x)
log2 x = log
log10 (2)
log A + log B = log(AB)
A
log A log B = log B
Complex
p
j=
1
j
e
= cos
j sin
j2
e
= j
cos = 21 ej + e j
sin = 2j1 ej
e j
a + jb = rej ; where r =
n
r ej
= rn ejn
r1 ej 1 r2 ej 2 = r1 r2 ej(
p
(a2 + b2 ) and
1+ 2)
= tan
1
b
a
1
1
b
a
b
a
and a 0
and a < 0
Exponential
ex = 1 + x +
x2
2!
+
x3
3!
+
x2
2
+
Logarithmic
loge (1 + x) = x
x3
3
where (jxj < 1)
Binomial
(1 + x)n = 1 + nx +
n(n 1) 2
x
2!
+
where jnxj < 1
+ (x
a)2
Taylor
f (x) = f (a)+(x
(x)
a) dfdx
x=a
1 d2 f (x)
2! d2 x
Derivatives and Integrals
d(au)
= a d(u)
dx
dx
d(u+v)
= d(u)
+ d(v)
dx
dx
dx
d(uv)
d(v)
= u dx + v d(u)
dx
dx
d(xn )
n 1
=
nx
dx
d(eu )
= eu d(u)
dx
dx
ax
d(e )
ax
=
ae
dx
d(sin u)
= cos u d(u)
dx
dx
d(cos u)
= sin u d(u)
dx
dx
d(sin ax)
=
a
cos
(ax)
dx
d(cos ax)
= a sin (ax)
dx
+
x=a
+(x
a)n
1 dn f (x)
n! dn x
x=a
d(tan )
= sec2
dx
d(cot )
= cosec2
dx
d(sec )
= tan sec
dx
d(cosec )
= cot cosec
dx
d(ln ax)
x)
= d(ln
= x1
dx
dx
Inde…nite Integrals
Z
Z
au dx = a u dx
Z
Z
Z
(u + v) dx = u dx + v dx
Z
Z
Z
Z
Z
xp dx =
1
x
xp+1
p+1
(p 6=
dx = ln jxj
u dv = uv
u
dv
dx
dx
Z
= uv
1)
v du
Z
v du
dx
dx
sin(ax) =
Z
1
a
cos(ax) =
1
a
sin2 (ax) =
Z
x
2
cos2 (ax) =
x
2
x sin(ax) =
sin(ax) ax cos(ax)
a2
x cos(ax) =
cos(ax)+ax sin(ax)
a2
Z
Z
Z
cos (ax)
sin (ax)
sin(2ax)
4a
+
sin(2ax)
4a
Z
x2 sin(ax) =
2ax sin(ax)+2 cos(ax) (ax)2 cos(ax)
a3
x2 cos(ax) =
Z
2ax cos(ax) 2 sin(ax)+(ax)2 sin(ax)
a3
eax dx =
Z
eax
a
Z
Z
Z
Z
Z
eax
a2
xeax dx =
(ax
1)
Z
xn eax
a
xn eax dx =
n
a
1 ax2
e
2a
2
xeax dx =
xn 1 eax dx
eax sin (bx) dx =
eax
a2 +b2
[a sin(bx)
eax cos (bx) dx =
eax
a2 +b2
[a cos(bx) + b sin(bx)]
Z
Z
Z
1
dx
(a2 +b2 x2 )
=
1
ab
x2
dx
(a2 +b2 x2 )
=
x
b2
x
dx
(a2 +x2 )
1
tan
a
b3
b cos(bx)]
bx
a
tan
1
bx
a
= 12 ln (x2 + a2 )
De…nite Integrals
Z1
0
x sin(ax)
dx
b2 +x2
Z1
cos(ax)
dx
b2 +x2
=
0
Z1
sinc dx =
2b
Z1
ax2
dx =
1
2
0
Z1
x2 e
ax2
ab)
where a > 0 and b > 0
sinc2 dx =
ax
p
dx =
0
Z1
xn e
0
e(
where a > 0 and b > 0
1
2
0
0
Z1
e
ab)
= 2 e(
dx =
a
1
4a
for a > 0
p
n!
an+1
a
for a > 0