FORMULA SHEET (TEST #1) Some identities 1. cos2(x) + sin2(x)=1

FORMULA SHEET (TEST #1)
Some identities
1. cos2 (x) + sin2 (x) = 1
1 + cos(2x)
2
1
−
cos(2x)
3. sin2 (x) =
2
2. cos2 (x) =
4. cosh2 (x) − sinh2 (x) = 1
cosh(2x) + 1
2
cosh(2x)
−1
6. sinh2 (x) =
2
5. cosh2 (x) =
Inverse Trigonometric Integral Identities
!
du
u
√
1.
= sin−1 ( ) + C, u2 < a2
a
a2 − u2
!
du
1
u
2.
= tan−1 ( ) + C
a2 + u2
a
a
" "
!
" "
du
1
−1 " u "
√
3.
= sec " " + C, u2 > a2
2
2
a
a
u u −a
Inverse Hyperbolic-Trig Integral Identities
!
du
u
√
1.
= sinh−1 ( ) + C, a > 0
a
a2 + u2
!
du
u
√
2.
= cosh−1 ( ) + C, u > a > 0
a
u2 − a2
!
du
1
u
3.
= tanh−1 ( ) + C, if u2 < a2
a2 − u2
a
a
!
du
1
u
4.
= coth−1 ( ) + C, if u2 > a2
a2 − u2
a
a
!
du
1
u
√
5.
= − sech−1 ( ) + C, 0 < u < a
2
2
a
a
u a −u
" "
!
"u"
du
1
√
6.
= − csch−1 "" "" + C, u #= 0
2
2
a
a
u a +u
Moments, Mass and Center of Mass of a Thin Rod along x-axis
! b
! b
MO
Mass: M =
δ(x)dx
Moments: MO =
xδ(x)dx
Center of Mass: x̄ =
M
a
a
Moments, Mass and Center of Mass of a Thin Plate
!
!
!
Mass: M = dm Moments: Mx = ỹ dm, My = x̃ dm
Center of Mass: x̄ =
My
Mx
, ȳ =
M
M