Examples of the Processes of the Differential and Integral Calculus |
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Page 252
... Sdx ( 21 ) fdx ( 22 ) d ( 23 ) -x ( x2- a2 ) x ( x2 + a2 ) 3 x dx ( x + a ) 3 x ( x − a ) 3 dx = = 1 + x ( 1 - x2 ) - = sin1 x − ( 1 − æ3 ) § . - ( x2 — a2 ) § 1 - a sec a ( x2 + a2 ) 3 + a log { ( x2 = sec √ ( x + a ) } + ( x ...
... Sdx ( 21 ) fdx ( 22 ) d ( 23 ) -x ( x2- a2 ) x ( x2 + a2 ) 3 x dx ( x + a ) 3 x ( x − a ) 3 dx = = 1 + x ( 1 - x2 ) - = sin1 x − ( 1 − æ3 ) § . - ( x2 — a2 ) § 1 - a sec a ( x2 + a2 ) 3 + a log { ( x2 = sec √ ( x + a ) } + ( x ...
Page 290
... ( Sdx + Tdy ) , d d = dx dy d R dS d S d l ' dx + dy 1 dx + dy . or dy dy dx dx But from the indeterminateness of dr , and dy this in- volves the conditions . d T = d.S d R d S dy da dx d2 R d2 Se d2 T Whence = = dy dx dy ' da and ...
... ( Sdx + Tdy ) , d d = dx dy d R dS d S d l ' dx + dy 1 dx + dy . or dy dy dx dx But from the indeterminateness of dr , and dy this in- volves the conditions . d T = d.S d R d S dy da dx d2 R d2 Se d2 T Whence = = dy dx dy ' da and ...
Page 378
... Sdx = 0 , - we can obtain three integrals , U = a , V = b , W = c , the integral of the proposed equation is U = ƒ ( V , W ) , or 4 ( U , V , W ) = 0 . ( 8 ) Let du du du ( u + y + x ) + ( u + x + x ) + ( u + x + y ) = x + y + ≈ . dx ...
... Sdx = 0 , - we can obtain three integrals , U = a , V = b , W = c , the integral of the proposed equation is U = ƒ ( V , W ) , or 4 ( U , V , W ) = 0 . ( 8 ) Let du du du ( u + y + x ) + ( u + x + x ) + ( u + x + y ) = x + y + ≈ . dx ...
Page 425
... sdx - ay - ¦ Sdæ { y ( 1 — — — ; ) * + b ( 1 - 2 ) sin ( 22 ) } 1 小 a2 b ( a2 x2 - Now fedy represents the area of a section parallel to the plane yx , and at a distance equal to : the integral must therefore be taken between the ...
... sdx - ay - ¦ Sdæ { y ( 1 — — — ; ) * + b ( 1 - 2 ) sin ( 22 ) } 1 小 a2 b ( a2 x2 - Now fedy represents the area of a section parallel to the plane yx , and at a distance equal to : the integral must therefore be taken between the ...
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