A Treatise on Infinitesimal Calculus ... |
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Page 118
... effected on the original integral . The following examples illustrate the method , and the development of it will be best understood by them . Ex . 1. Evaluate u = ୮ " Take the r - differential : then du dr = = 0 e ar - -e - bx Ꮖ -bx ...
... effected on the original integral . The following examples illustrate the method , and the development of it will be best understood by them . Ex . 1. Evaluate u = ୮ " Take the r - differential : then du dr = = 0 e ar - -e - bx Ꮖ -bx ...
Page 123
... effecting in each member the integration which stands first , we have ƒˆ ˆ { ƒ ' ( x + y » √ = 1 ) −ƒ ' ( x + y ̧ √ − 1 ) } d.x Yn = √ = 1 { " " { ƒ ̃ ( xn + y√ = 1 ) −ƒ ′ ( x + y√ = 1 ) } dy . ( 127 ) 0 Let us apply this ...
... effecting in each member the integration which stands first , we have ƒˆ ˆ { ƒ ' ( x + y » √ = 1 ) −ƒ ' ( x + y ̧ √ − 1 ) } d.x Yn = √ = 1 { " " { ƒ ̃ ( xn + y√ = 1 ) −ƒ ′ ( x + y√ = 1 ) } dy . ( 127 ) 0 Let us apply this ...
Page 125
... effecting in each member the integration which stands first , we have [ * * xo = = ' { ( a + y „ √ − 1 ) ƒ ' ( ax + xy „ √ − 1 ) / = 1 / " " { x n ƒ ' ( ax „ + x „ ¥ √ −1 ) yo - ( a + y。√1 ) ƒ ' ( ax + xy√ − 1 ) } dx −x。ƒ ...
... effecting in each member the integration which stands first , we have [ * * xo = = ' { ( a + y „ √ − 1 ) ƒ ' ( ax + xy „ √ − 1 ) / = 1 / " " { x n ƒ ' ( ax „ + x „ ¥ √ −1 ) yo - ( a + y。√1 ) ƒ ' ( ax + xy√ − 1 ) } dx −x。ƒ ...
Page 152
... effected ; hereby a new series is formed , which is the value of the given integral . If the sum of this series can be expressed in general and finite terms , that sum is the value of the definite integral ; but when this cannot be ...
... effected ; hereby a new series is formed , which is the value of the given integral . If the sum of this series can be expressed in general and finite terms , that sum is the value of the definite integral ; but when this cannot be ...
Page 244
... effected by a comparison of the given series , with other series which are known to be convergent or divergent . Let there be two series U = U1 + U2 + Uz + ... V = V2 + Vq + Vg + ... ; ( 21 ) ( 22 ) the ratio of the nth terms of which ...
... effected by a comparison of the given series , with other series which are known to be convergent or divergent . Let there be two series U = U1 + U2 + Uz + ... V = V2 + Vq + Vg + ... ; ( 21 ) ( 22 ) the ratio of the nth terms of which ...
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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