A Treatise on Infinitesimal Calculus ... |
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Page 5
... similar to EH , contained between the two ordinates MP and NQ . Now as MP = f ( x ) , NQ = f ( x + dx ) , dx MN = dx ; consequently MPQN = 2 = f ( x ) dx , since da is infinitesimal ; and the whole area is the sum of all the similar ...
... similar to EH , contained between the two ordinates MP and NQ . Now as MP = f ( x ) , NQ = f ( x + dx ) , dx MN = dx ; consequently MPQN = 2 = f ( x ) dx , since da is infinitesimal ; and the whole area is the sum of all the similar ...
Page 20
... dx dx n x dx a2 + x2 ' + log ( a2 + x2 ) . , 2 and of similar forms . 15. ] Integration of Since x2 - 1 a2 x2- a2 › a2 — x2 11 1 = ; 2 a X- a x + a therefore dx - a2 = = = 1 2 dx 20 [ 15 . INTEGRATION OF ALGEBRAICAL FUNCTIONS .
... dx dx n x dx a2 + x2 ' + log ( a2 + x2 ) . , 2 and of similar forms . 15. ] Integration of Since x2 - 1 a2 x2- a2 › a2 — x2 11 1 = ; 2 a X- a x + a therefore dx - a2 = = = 1 2 dx 20 [ 15 . INTEGRATION OF ALGEBRAICAL FUNCTIONS .
Page 25
... similar to that which is applied to one set must be applied to each of the others . 19. ] Let all the roots of ƒ ( x ) be unequal , so that Then F ( x ) f ( x ) f ( x ) = · ( " v — x ) · · · · · · ( 3 v — x ) ( ' n — x ) ( 22 ) may be ...
... similar to that which is applied to one set must be applied to each of the others . 19. ] Let all the roots of ƒ ( x ) be unequal , so that Then F ( x ) f ( x ) f ( x ) = · ( " v — x ) · · · · · · ( 3 v — x ) ( ' n — x ) ( 22 ) may be ...
Page 38
... 1 sin n π n π π COS -V - 1 sin n n - - ( π COS + √ - 1 sin X- COS √1 sin n π - n π 2 x cos 2 which becomes - 1 n x2- 2 x cos n FIS π +1 and the other pairs give similar results ; so that 38 [ 24 . INTEGRATION OF RATIONAL FRACTIONS .
... 1 sin n π n π π COS -V - 1 sin n n - - ( π COS + √ - 1 sin X- COS √1 sin n π - n π 2 x cos 2 which becomes - 1 n x2- 2 x cos n FIS π +1 and the other pairs give similar results ; so that 38 [ 24 . INTEGRATION OF RATIONAL FRACTIONS .
Page 39
... similar to that employed when n is even , the last pair becomes - n . 2 2 x cos π - 2 1 n n n -2 x2 - 2x cos π + 1 n and therefore , when n is odd , π -- x cos n dx √ 4 = 1 / ( 2000 - 2 ) dr dx x ” +1 = - x2 - 2x cos 2x cos π n +1 n ...
... similar to that employed when n is even , the last pair becomes - n . 2 2 x cos π - 2 1 n n n -2 x2 - 2x cos π + 1 n and therefore , when n is odd , π -- x cos n dx √ 4 = 1 / ( 2000 - 2 ) dr dx x ” +1 = - x2 - 2x cos 2x cos π n +1 n ...
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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