Treatise On the Mechanics of Engineering And Machinery With Requisite ...

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Treatise On the Mechanics of Engineering And Machinery With Requisite ...
Julius Ludwig Weisbach
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x) = ^d (nat. log. x) = ^ • — .
X 2. 8 nat. log. (^-^) = 8 [log. (2 + x) — log. x''] = 8 log. (2 + a:) — 8 log. (x^) — 8j? _ o 8^ __ _ (4 + ^) 8^ ^2 + x' X ~~ a? (2 + a;)' 3. 8 {nat. log. --~^ = 8 [.nat. log. (e^ — 1)]— 8 [nat. log. (e^+ 1)] _ 8 (e^) 8(g^) _ e^dx _ e^dx _ 2 e' 8 a ?
~ ~e^^^ ~ e*+ 1 ~" e^ — 1 e^ + 1 "~ e'^* — 1 ' Art. 22. If the differential formulae of the foregoing article be- come inverted, we meet with other important integral formulae as follows.
[Art. 23. ELEMENTS OF ANALY
...SIS. 33 From 8 (a^) = , there follows I = a^, i. e.: I. f a^ dx = 7na^ = a^ \ naL log. a, and hence: II. fe^dx = e^.
Further, from 8 (log.^x) = , there follows I = log.^x^ 1. e.: rdx 1 — = — log, X = naL log. x. and the formula 8 (nat. log.
x)= — gives the same.
From this, the following examples may be easily performed: y^sx 1 g^ ^ ife'^"' 8 (5 a; — 1) = J J^^-^ =fxdx+fdx + 2f^-^-^^ = ^ + x+2naLlog.(x — l).
x'^dx = — ----, leaves the last integral undetermined; for if we put m = — 1, there follows: J8j7 C x^ — ■' = I x"^ 8^7 = — -• -f a constant = oo + constant; but if we a; •/ put X = I -{- u, and dx = 8w, we obtain: — = = (1 — u + u^ — u^ 4- u* ) du.


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