1 cos(2x) = cos2(x)

FORMULA SHEET
Some identities
cos2 (x) + sin2 (x) = 1
cos(2x) = cos2 (x) − sin2 (x)
sin(2x) = 2 sin(x) cos(x)
1 + cos(2x)
cos2 (x) =
2
1 − cos(2x)
2
sin (x) =
2
2
Volume of a Solid of Revolution
Disk Method: ∆V = πR2 ∆(·)
Washer Method: ∆V = π[R2 − r2 ]∆(·)
Shell Method: ∆V = 2πrh∆(·)
Arclength of curve C/Surface Area
If C : y = f (x) ≥ 0, a ≤ x ≤ b, then
L=
Z bp
2
cosh (x) − sinh (x) = 1
cosh(2x) + 1
cosh2 (x) =
2
cosh(2x) − 1
2
sinh (x) =
2
Inverse Trigonometric Integral Identities
Z
du
u
√
= sin−1 ( ) + C, u2 < a2
a
a2 − u2
Z
1
u
du
= tan−1 ( ) + C
2
2
a
a Z a +u
du
1
−1 u √
= sec + C, u2 > a2
a
a
u u2 − a2
Inverse Hyperbolic-Trig Integral Identities
Z
u
du
√
= sinh−1 ( ) + C, a > 0
2
2
a
a +u
Z
du
−1 u
√
= cosh ( ) + C, u > a > 0
a
u2 − a2
Z
du
1
u
= tanh−1 ( ) + C, if u2 < a2
2
2
a
a
Z a −u
du
1
−1 u
= coth ( ) + C, if u2 > a2
2 − u2
a
a
a
Z
u
du
1
√
= − sech−1 ( ) + C, 0 < u < a
2
2
a
a
Z u a −u
u
1
du
√
= − csch−1 + C, u 6= 0
a
a
u a2 + u2
Moments, Mass and Center of Mass of a
Thin Rod along x-axis
Z
b
Mass: M =
δ(x)dx
Z b
Moments: MO =
xδ(x)dx
a
a
Center of Mass: x̄ = MO /M
Moments, Mass and Center of Mass of a
Thin Plate
Z
Mass: M = dm,
Z
Z
Moments: Mx = ỹ dm, My = x̃ dm
Center of Mass: x̄ = My /M, ȳ = Mx /M
1 + f 0 (x)2 dx
a
and surface area of surface generated by rotating
curve C about the x-axis is
Z
SA =
b
p
2πf (x) 1 + f 0 (x)2 dx.
a
If C : x = g(y) ≥ 0, a ≤ y ≤ b, then
L=
Z bp
1 + g 0 (y)2 dy
a
and surface area of surface generated by rotating
curve C about the y-axis is
Z b
p
SA =
2πg(y) 1 + g 0 (y)2 dy.
a
Some useful limits
ln n
=0
n→∞ n
lim
lim x1/n
n→∞ lim
n→∞
= 1, x > 0
x n
1+
= ex , any x
n
lim
n→∞
√
n
n=1
lim xn = 0, |x| < 1
xn
lim
= 0, any x
n→∞ n!
n→∞