A Treatise on Infinitesimal Calculus ... |
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Page 7
... Hence it follows that the definite integral of F ' ( x ) dx between the limits x and x is the value of the indefinite integral when x = x , „ , less its value when x = x ; on this account it is fre- quently and conveniently expressed as ...
... Hence it follows that the definite integral of F ' ( x ) dx between the limits x and x is the value of the indefinite integral when x = x , „ , less its value when x = x ; on this account it is fre- quently and conveniently expressed as ...
Page 8
... Hence we have finally , subject to the condition that F ' ( x ) is finite and continuous within the limits x , and a 。, [ ** ' x ' ( x ) dx = - F ′ ( x ̧ ) ( X1 − x ̧q ) + F ′ ( X1 ) ( X2 −X1 ) + ... .. ... + F ′ ( xn− 1 ) ( x n ...
... Hence we have finally , subject to the condition that F ' ( x ) is finite and continuous within the limits x , and a 。, [ ** ' x ' ( x ) dx = - F ′ ( x ̧ ) ( X1 − x ̧q ) + F ′ ( X1 ) ( X2 −X1 ) + ... .. ... + F ′ ( xn− 1 ) ( x n ...
Page 9
... Hence also Ja and d are symbols of operations which destroy each other ; and consequently = do = Sa = a f 10 = f ' = 1 ; d unity being used as a symbol of an operation which operating on a function leaves it unaltered . Hence according ...
... Hence also Ja and d are symbols of operations which destroy each other ; and consequently = do = Sa = a f 10 = f ' = 1 ; d unity being used as a symbol of an operation which operating on a function leaves it unaltered . Hence according ...
Page 11
... Hence , and by means of the former theorem , ... ( 19 ) [ " { ¥ ' ( x ) + √ = 1ƒ ' ( x ) } dx = [ * x ( x ) dx + √ = 1 [ * ƒ ′ ( x ) dx . ( 20 ) dæ THEOREM III . - If the infinitesimal element is of the form f ( x ) × v ′ ( x ) dx ...
... Hence , and by means of the former theorem , ... ( 19 ) [ " { ¥ ' ( x ) + √ = 1ƒ ' ( x ) } dx = [ * x ( x ) dx + √ = 1 [ * ƒ ′ ( x ) dx . ( 20 ) dæ THEOREM III . - If the infinitesimal element is of the form f ( x ) × v ′ ( x ) dx ...
Page 14
... Hence we have xo e dx = en- € * 0 . Ex . 5. Determine cos x dx . Let x - xo be divided into n equal elements , each of which is equal to i ; so that xo x - xo = ni ; cos x dx = i cosx + icos ( x + i ) + + icos { x + ( n - 1 ) i } i ...
... Hence we have xo e dx = en- € * 0 . Ex . 5. Determine cos x dx . Let x - xo be divided into n equal elements , each of which is equal to i ; so that xo x - xo = ni ; cos x dx = i cosx + icos ( x + i ) + + icos { x + ( n - 1 ) i } i ...
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A Treatise on Infinitesimal Calculus: Containing Differential and Integral ... Bartholomew Price No preview available - 2015 |
Common terms and phrases
a₁ a₂ angle application axis Beta-function bx dx consequently convergent series coordinates cosec cx² cycloid definite integral denoted determined differential double integral dx a² dx dx dx dy dx Ex dx² dy dx dy² e-ax element-function ellipse equal evaluation expressed find the area finite and continuous fraction function Gamma-function geometrical given Hence infinite infinitesimal infinitesimal element Integral Calculus intrinsic equation involute left-hand member length let us suppose limits of integration multiple integrals plane curve polar coordinates preceding proper fraction radius range of integration replaced result right-hand member subject-variable substituting surface symbols theorem tion values variable x-integration x₁ x²)¹ x²)³ αξ πα