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712 ESSENTIAL FORMULAE
Integral Calculus
Standard integrals
y!
y dx
axn axn+1
n + 1+ c
(except where n = !1)
cos ax1a
sin ax + c
sin ax !1a
cos ax + c
sec2 ax1a
tan ax + c
cosec2 ax !1a
cot ax + c
cosec ax cot ax !1a
cosec ax + c
sec ax tan ax1a
sec ax + c
eax 1a
eax + c
1x
ln x + c
tan ax1a
ln ( sec ax) + c
cos2 x12
"x + sin 2x
2
#+ c
sin2 x12
"x ! sin 2x
2
#+ c
tan2 x tan x ! x + c
cot2 x !cot x ! x + c
1$
(a2 ! x2)sin!1 x
a+ c
$(a2 ! x2)
a2
2sin!1 x
a+ x
2
$(a2 ! x2) + c
1(a2 + x2)
1a
tan!1 xa
+ c
y!
y dx
1$
(x2 + a2)sinh!1 x
a+ c or
ln
%x +
$(x2 + a2)a
&
+ c
$(x2 + a2)
a2
2sinh!1 x
a+ x
2
$(x2 + a2) + c
1$
(x2 ! a2)cosh!1 x
a+ c or
ln
%x +
$(x2 ! a2)a
&
+ c
$(x2 ! a2)
x2
$(x2 ! a2) ! a2
2cosh!1 x
a+ c
t = tan!
2substitution
To determine! 1
a cos ! + b sin ! + cd! let
sin ! = 2t(1 + t2)
cos ! = 1 ! t2
1 + t2 and
d! = 2 dt(1 + t2)
Integration by parts
If u and v are both functions of x then:
'u
dv
dxdx = uv !
'v
dudx
dx
Reduction formulae'
xnex dx = In = xnex ! nIn!1'
xn cos x dx = In = xn sin x + nxn!1 cos x
!n(n ! 1)In!2
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ESSENTIAL FORMULAE 713
! !
0xn cos x dx = In = !n!n!1 ! n(n ! 1)In!2
!xn sin x dx = In = !xn cos x + nxn!1 sin x
!n(n ! 1)In!2!
sinn x dx = In = !1n
sinn!1 x cos x + n ! 1n
In!2
!cosn x dx = In = 1
ncosn!1 sin x + n ! 1
nIn!2
! !/2
0sinn x dx =
! !/2
0cosn x dx = In = n ! 1
nIn!2
!tann x dx = In = tann!1 x
n ! 1! In!2
!(ln x)n dx = In = x( ln x)n ! nIn!1
With reference to Fig. FA4.
0 x 5 a x 5 b x
y
y 5 f (x)
A
Figure FA4
Area under a curve:
area A =! b
ay dx
Mean value:
mean value = 1b ! a
! b
ay dx
R.m.s. value:
r.m.s. value =
"##$%
1b ! a
! b
ay2 dx
&
Volume of solid of revolution:
volume =! b
a!y2 dx about the x-axis
Centroids
With reference to Fig. FA5:
x =
! b
axy dx
! b
ay dx
and y =12
! b
ay2 dx
! b
ay dx
Area A
y 5 f (x)
C
yx
0 x 5 a x 5 b x
y
Figure FA5
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714 ESSENTIAL FORMULAE
Second moment of area and radius of gyration
Shape Position of axis Second moment Radius ofof area, I gyration, k
Rectangle (1) Coinciding with bbl3
31!3length l
(2) Coinciding with llb3
3b!3
breadth b
(3) Through centroid,bl3
121!12parallel to b
(4) Through centroid,lb3
12b!12parallel to l
Triangle (1) Coinciding with bbh3
12h!6Perpendicular
(2) Through centroid,bh3
36h!18
height h
parallel to basebase b
(3) Through vertex,bh3
4h!2parallel to base
Circle (1) Through centre,!r4
2r!2radius r perpendicular to plane
(i.e. polar axis)
(2) Coinciding with diameter!r4
4r2
(3) About a tangent5!r4
4
!5
2r
Semicircle Coinciding with!r4
8r2radius r diameter
Theorem of Pappus
With reference to Fig. FA5, when the curve is rotatedone revolution about the x-axis between the limitsx = a and x = b, the volume V generated is given by:V = 2!Ay.
Parallel axis theorem:
If C is the centroid of area A in Fig. FA6 then
Ak2BB = Ak2
GG + Ad2 or k2BB = k2
GG + d2
G B
CArea A
d
G B
Figure FA6
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ESSENTIAL FORMULAE 715
Perpendicular axis theorem:
If OX and OY lie in the plane of area A in Fig. FA7,
then Ak2OZ = Ak2
OX + Ak2OY or k2
OZ = k2OX + k2
OY
Z
Area A
O
X
Y
Figure FA7
Numerical integration
Trapezoidal rule
!ydx !
"width ofinterval
# $12
"first + lastordinates
#
+"
sum of remainingordinates
#%
Mid-ordinate rule!
ydx !"
width ofinterval
# "sum of
mid-ordinates
#
Simpson’s rule
!ydx ! 1
3
"width ofinterval
# $"first + lastordinate
#
+ 4"
sum of evenordinates
#
+ 2"
sum of remainingodd ordinates
#%
Differential Equations
First order differential equations
Separation of variables
Ifdydx
= f (x) then y =!
f (x) dx
Ifdydx
= f (y) then!
dx =!
dyf (y)
Ifdydx
= f (x) · f (y) then!
dyf (y)
=!
f (x) dx
Homogeneous equations
If Pdydx
= Q, where P and Q are functions of bothx and y of the same degree throughout (i.e. ahomogeneous first order differential equation) then:
(i) Rearrange Pdydx
= Q into the formdydx
= QP
(ii) Make the substitution y = vx (where v is afunction of x), from which, by the product rule,dydx
= v(1) + xdv
dx
(iii) Substitute for both y anddydx
in the equationdydx
= QP
(iv) Simplify, by cancelling, and then separate the
variables and solve using thedydx
= f (x) · f (y)method
(v) Substitute v = yx
to solve in terms of the originalvariables.
Linear first order
Ifdydx
+ Py = Q, where P and Q are functions ofx only (i.e. a linear first order differential equation),then
(i) determine the integrating factor, e&
P dx
(ii) substitute the integrating factor (I.F.) intothe equation
y (I.F.) =!
(I.F.) Q dx
(iii) determine the integral&
(I.F.)Q dx
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716 ESSENTIAL FORMULAE
Numerical solutions of first order differentialequations
Euler’s method: y1 = y0 + h(y!)0
Euler-Cauchy method: yP1 = y0 + h(y!)0
and yC1 = y0 + 12
h[(y!)0 + f (x1, yp1 )]
Runge-Kutta method:
To solve the differential equationdydx
= f (x, y) giventhe initial condition y = y0 at x = x0 for a range ofvalues of x = x0(h)xn:
1. Identify x0, y0 and h, and values of x1, x2, x3, . . .
2. Evaluate k1 = f (xn, yn) starting with n = 0
3. Evaluate k2 = f!
xn + h2
, yn + h2
k1
"
4. Evaluate k3 = f!
xn + h2
, yn + h2
k2
"
5. Evaluate k4 = f(xn + h, yn + hk3)
6. Use the values determined from steps 2 to 5 toevaluate:
yn+1 = yn + h6{k1 + 2k2 + 2k3 + k4}
7. Repeat steps 2 to 6 for n = 1, 2, 3, . . .
Second order differential equations
If ad2ydx2 + b
dydx
+ cy = 0 (where a, b and c areconstants) then:
(i) rewrite the differential equation as(aD2 + bD + c)y = 0
(ii) substitute m for D and solve the auxiliaryequation am2 + bm + c = 0
(iii) if the roots of the auxiliary equation are:
(a) real and different, say m = ! and m = "then the general solution is
y = Ae!x + Be"x
(b) real and equal, say m = ! twice, then thegeneral solution is
y = (Ax + B)e!x
(c) complex, say m = ! ± j", then the generalsolution is
y = e!x(A cos "x + B sin "x)
(iv) given boundary conditions, constants A and Bcan be determined and the particular solutionobtained.
If ad2ydx2 + b
dydx
+ cy = f (x) then:
(i) rewrite the differential equation as(aD2 + bD + c)y = 0.
(ii) substitute m for D and solve the auxiliaryequation am2 + bm + c = 0.
(iii) obtain the complimentary function (C.F.), u, asper (iii) above.
(iv) to find the particular integral, v, first assume aparticular integral which is suggested by f (x),but which contains undetermined coefficients(See Table 51.1, page 482 for guidance).
(v) substitute the suggested particular integral intothe original differential equation and equaterelevant coefficients to find the constantsintroduced.
(vi) the general solution is given by y = u + v.(vii) given boundary conditions, arbitrary constants
in the C.F. can be determined and the particularsolution obtained.
Higher derivatives
y y(n)
eax an eax
sin ax an sin#
ax + n#
2
$
cos ax an cos#
ax + n#
2
$
xa a!(a " n)!x
a"n
sinh axan
2{[1 + ("1)n] sinh ax
+ [1 " ("1)n] cosh ax}
cosh axan
2{[1 " ("1)n] sinh ax
+[1 + ("1)n] cosh ax}
ln ax ("1)n"1 (n " 1)!xn
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ESSENTIAL FORMULAE 717
Leibniz’s theorem
To find the n’th derivative of a product y = uv:
y(n) = (uv)(n) = u(n)v + nu(n!1)v(1)
+n(n ! 1)2! u(n!2)v(2)
+n(n ! 1)(n ! 2)3! u(n!3)v(3) + · · ·
Power series solutions of second order differentialequations.
(a) Leibniz-Maclaurin method
(i) Differentiate the given equation n times,using the Leibniz theorem,
(ii) rearrange the result to obtain the recurrencerelation at x = 0,
(iii) determine the values of the derivatives atx = 0, i.e. find (y)0 and (y")0,
(iv) substitute in the Maclaurin expansion fory = f (x),
(v) simplify the result where possible and applyboundary condition (if given).
(b) Frobenius method
(i) Assume a trial solution of the form:y = xc{a0 + a1x + a2x2 + a3x3 + · · · +
arxr + · · · } a0 #= 0,
(ii) differentiate the trial series to find y"and y"",
(iii) substitute the results in the given differentialequation,
(iv) equate coefficients of corresponding pow-ers of the variable on each side of theequation: this enables index c and coeffi-cients a1, a2, a3, . . . from the trial solution,to be determined.
Bessel’s equation
The solution of x2 d2ydx2 + x
dydx
+ (x2 ! v2)y = 0
is:
y = Axv
!1 ! x2
22(v + 1)
+ x4
24 $ 2!(v + 1)(v + 2)
! x6
26 $ 3!(v + 1)(v + 2)(v + 3)+ · · ·
"
+ Bx!v
!1 + x2
22(v ! 1)+ x4
24 $ 2!(v ! 1)(v ! 2)
+ x6
26 $ 3!(v ! 1)(v ! 2)(v ! 3)+ · · ·
"
or, in terms of Bessel functions and gammafunctions:
y = AJv(x) + BJ!v(x)
= A#x
2
$v!
1!(v + 1)
! x2
22(1!)!(v + 2)
+ x4
24(2!)!(v + 4)! · · ·
"
+ B#x
2
$!v!
1!(1 ! v)
! x2
22(1!)!(2 ! v)
+ x4
24(2!)!(3 ! v)! · · ·
"
In general terms:
Jv(x) =#x
2
$v%%
k=0
(!1)kx2k
22k(k!)!(v + k + 1)
and J!v(x) =#x
2
$!v%%
k=0
(!1)kx2k
22k(k!)!(k ! v + 1)
and in particular:
Jn(x) =#x
2
$n!
1n! ! 1
(n + 1)!#x
2
$2
+ 1(2!)(n + 2)!
#x2
$4! · · ·
"
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718 ESSENTIAL FORMULAE
J0(x) = 1 ! x2
22(1!)2 + x4
24(2!)2
! x6
26(3!)2 + · · ·
and J1(x) = x2
! x3
23(1!)(2!) + x5
25(2!)(3!)
! x7
27(3!)(4!) + · · ·
Legendre’s equation
The solution of (1!x2)d2ydx2 !2x
dydx
+k(k+1)y = 0
is:
y = a0
!1 ! k(k + 1)
2! x2
+ k(k + 1)(k ! 2)(k + 3)4! x4 ! · · ·
"
+ a1
!x ! (k ! 1)(k + 2)
3! x3
+ (k ! 1)(k ! 3)(k + 2)(k + 4)5! x5 ! · · ·
"
Rodrigue’s formula
Pn(x) = 12nn!
dn(x2 ! 1)n
dxn
Statistics and Probability
Mean, median, mode and standard deviation
If x = variate and f = frequency then:
mean x =#
fx#f
The median is the middle term of a ranked set ofdata.The mode is the most commonly occurring value ina set of data.
Standard deviation
! =
$%%&'# (
f (x ! x)2)
#f
*
for a population
Binomial probability distribution
If n = number in sample, p = probability of theoccurrence of an event and q = 1 ! p, then theprobability of 0, 1, 2, 3, . . . occurrences is given by:
qn, nqn!1p,n(n ! 1)
2! qn!2p2,
n(n ! 1)(n ! 2)3! qn!3p3, . . .
(i.e. successive terms of the (q + p)n expansion).
Normal approximation to a binomial distribution:
Mean = np Standard deviation ! = "(npq)
Poisson distribution
If " is the expectation of the occurrence of an eventthen the probability of 0, 1, 2, 3, . . . occurrences isgiven by:
e!", "e!", "2 e!"
2! , "3 e!"
3! , . . .
Product-moment formula for the linear correlationcoefficient
Coefficient of correlation r =#
xy+,-#
x2. -#
y2./
where x = X ! X and y = Y ! Y and (X1, Y1),(X2, Y2), . . . denote a random sample from a bivari-ate normal distribution and X and Y are the meansof the X and Y values respectively.
Normal probability distribution
Partial areas under the standardized normal curve —see Table 58.1 on page 561.
Student’s t distribution
Percentile values (tp) for Student’s t distribution with# degrees of freedom — see Table 61.2 on page 587.
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ESSENTIAL FORMULAE 719
Chi-square distribution
Percentile values (!2p) for the Chi-square distribu-
tion with " degrees of freedom—see Table 63.1 onpage 609.
!2 = ! "(o ! e)2
e
#where o and e are the observed
and expected frequencies.
Symbols:
Population
number of members Np, mean µ, standard devia-tion #.
Sample
number of members N , mean x, standard deviation s.
Sampling distributions
mean of sampling distribution of means µxstandard error of means #xstandard error of the standard deviations #s.
Standard error of the means
Standard error of the means of a sample distribu-tion, i.e. the standard deviation of the means ofsamples, is:
#x = #"N
$%Np ! NNp ! 1
&
for a finite population and/or for sampling withoutreplacement, and
#x = #"N
for an infinite population and/or for sampling withreplacement.
The relationship between sample mean andpopulation mean
µx = µ for all possible samples of size N are drawnfrom a population of size Np.
Estimating the mean of a population (! known)
The confidence coefficient for a large sample size,(N # 30) is zc where:
Confidence Confidencelevel % coefficient zc
99 2.5898 2.3396 2.0595 1.9690 1.64580 1.2850 0.6745
The confidence limits of a population mean basedon sample data are given by:
x ± zc#"N
$%Np ! NNp ! 1
&
for a finite population of size Np, and by
x ± zc#"N
for an infinite population
Estimating the mean of a population (! unknown)
The confidence limits of a population mean basedon sample data are given by: µx ± zc#x.
Estimating the standard deviation of a population
The confidence limits of the standard deviation of apopulation based on sample data are given by:s ± zc#s.
Estimating the mean of a population based on asmall sample size
The confidence coefficient for a small sample size(N < 30) is tc which can be determined usingTable 61.1, page 582. The confidence limits of apopulation mean based on sample data is given by:
x ± tcs"(N ! 1)
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720 ESSENTIAL FORMULAE
Laplace Transforms
Function Laplace transformsf (t) L{f (t)} =
! !0 e"st f (t) dt
1 1s
k ks
eat 1s"a
sin at as2+a2
cos at ss2+a2
t 1s2
tn(n = positve integer) n!sn+1
cosh at ss2"a2
sinh at as2"a2
e"attn n!(s+a)n+1
e"at sin !t !(s+a)2+!2
e"at cos !t s+a(s+a)2+!2
e"at cosh !t s+a(s+a)2"!2
e"at sinh !t !(s+a)2"!2
The Laplace transforms of derivatives
First derivative
L"
dydx
#= sL{y} ! y(0)
where y(0) is the value of y at x = 0.
Second derivative
L"
dydx
#= s2L{y} ! sy(0) ! y"(0)
where y#(0) is the value ofdydx
at x = 0.
Fourier Series
If f (x) is a periodic function of period 2" then itsFourier series is given by:
f (x) = a0 +!$
n=1
(an cos nx + bn sin nx)
where, for the range "" to +":
a0 = 12"
% "
""f (x) dx
an = 1"
% "
""f (x) cos nx dx (n = 1, 2, 3, . . . )
bn = 1"
% "
""f (x) sin nx dx (n = 1, 2, 3, . . . )
If f (x) is a periodic function of period L then itsFourier series is given by:
f (x) = a0 +#$
n=1
&an cos
'2!nx
L
(+ bn sin
) 2!nxL
*+
where for the range "L2
to +L2
:
a0 = 1L
% L/2
"L/2f (x) dx
an = 2L
% L/2
"L/2f (x) cos
) 2"nxL
*dx (n = 1, 2, 3, . . . )
bn = 2L
% L/2
"L/2f (x) sin
) 2"nxL
*dx (n = 1, 2, 3, . . . )
Complex or exponential Fourier series
f (x) =!$
n="!cne j 2"nx
L
where cn = 1L
% L2
" L2
f (x)e"j 2"nxL dx
For even symmetry,
cn = 2L
% L2
0f (x) cos
) 2"nxL
*dx
For odd symmetry,
cn = "j2L
% L2
0f (x) sin
) 2"nxL
*dx