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prove that integration of 1/x is logx. how should i prove that..

prove that integration of 1/x is logx.
 
how should i prove that..

Grade:12th pass

2 Answers

parchan
13 Points
5 years ago
this is based on knowledge of differntiaton.as 1/x is of the form of x**-1.differntiation in a nutshell is the opposite of integration. so as d/dx of ln(x) is 1/x.hence integral of 1/x is ln(x).
idk how to prove it
get rekt
Ritika Das
105 Points
4 years ago
(ln(x))' = lim(d->0) [ ln(x+d) - ln(x) ] / d = lim ln((x+d)/x) / d
= lim (1/d) ln(1 + d/x) = lim [ ln (1 + d/x)(1/d) ].
Set u=d/x and substitute:
lim(u->0) [ ln (1 + u)(1/(ux)) ] = 1/x ln [ lim(u->0) (1 + u)(1/u) ]
= 1/x ln (e) 
and since ln(e) = 1
(ln(x))' = 1/x.
 and since integration is just the reverse of differentiation, then the integration of 1/x is simply ln(x)

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