By Noel M. Morris

A pocket reference of crucial formulae protecting: digital and electric engineering, measurements and keep watch over, common sense, telecommunications and arithmetic. Of worth to scholars at either BTEC nationwide and better point, in addition to at undergraduate point, specifically these learning digital and electric engineering.

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For every quadratic factor (ax 2 + bx + c) of M(x) corresponding partial fraction _P_ + ( In x X 1. For every linear factor (ax + b) of M(x) there is a --=----=--- + -------:: ax 2 + bx + c (ax 2 + bx + c) 2 X sin -t - (a2 - x2 )~ a I -tan -t I x2 + a2 cosh x sinh x sech 2 x a sinh x cosh x tanh x a Integration by parts fu dxdv 'Cover-up' rule If X dx = uv - fv dudx dx f(x) = p(x) (x +a)(x + b)(x +c)··· then the numerators of the separate fractions due to the factors (x +a), (x + b), etc, are determined by 'covering up' each of the factors in turn and evaluating the remainder of the expression by replacing each x term by the value of x which makes the 'covered up' factor zero.

Sin(ax+b) I --cos (ax+ b) cos(ax+b) I -sin (ax+ b) a a q b) 2 ax+ 4. For every repeated quadratic factor (ax 2 + bx + c) 2 in M(x) there is a corresponding partial fraction px + q rx + s a tan x -cot x cosec 2 x I px + q • 2 ax + bx +c 3. For every repeated factor (ax + b ) 2 of M(x) there is a · a correspond"mg par t"Ial f rae t"wn t h ere IS ax+b I -In (ax+ b) ax+ b sec 2 x ax +b 2. For every quadratic factor (ax 2 + bx + c) of M(x) corresponding partial fraction _P_ + ( In x X 1. For every linear factor (ax + b) of M(x) there is a --=----=--- + -------:: ax 2 + bx + c (ax 2 + bx + c) 2 X sin -t - (a2 - x2 )~ a I -tan -t I x2 + a2 cosh x sinh x sech 2 x a sinh x cosh x tanh x a Integration by parts fu dxdv 'Cover-up' rule If X dx = uv - fv dudx dx f(x) = p(x) (x +a)(x + b)(x +c)··· then the numerators of the separate fractions due to the factors (x +a), (x + b), etc, are determined by 'covering up' each of the factors in turn and evaluating the remainder of the expression by replacing each x term by the value of x which makes the 'covered up' factor zero.

24 dy d 2y If at a point p on a curve dx = 0 and dx 2 = 0, the curve may If f(D) = aoD(n) + atD(n-l) + ... + arD(n-r) + ... +an where ar is a constant and n is a positive integer, then either have a maximum point, or a minimum point, or a d2y point of inflection. If dx 2 changes sign as x changes from f(D)eax = eaxf(a) 1 1 eax = - eax f(D) f(a) (p - 8x) to (p + 8x ), then there is a point of inflection at p. f(D){eaxf(x)} = eaxf(D + a)f(x) 1 The total differential If z = f(x, y ), and both x andy are independent variables, then az ax 1 f(D) {eax f(x)} = eax f(D + a) f(x) 1 1 f(D 2 ) sin ax = f( _ a2 ) sin ax az ay dz = - dx +- dy INTEGRAL CALCULUS [f(a) =I= 0] - 1 1 f(D 2 ) cos ax = f( _ a2 ) cos ax [f(a) =I= 0] PARTIAL FRACTIONS Standard integrals A(x) y If f(x) = -(-, where A(x) and M(x) are polynomials in x, Mx) fy dx the degree of A(x) being less than that of M(x ), then kxn+I kxn - - (n =I= n+ I I) 1 k -eax a keax corresponding partial fraction - - .