Find the error in this proof of 0=1:
∫(1/x) dx
= ∫(1/x)*(1) dx (Mult. Identity)
= (1/x) x - ∫(-1/x^2)*x dx (Integ by Parts)
= 1 + ∫(1/x) dx (Simplify)
Hence, ∫(1/x) dx = 1 + ∫(1/x) dx, therefore 0 = 1.
(In reply to
Solution by np_rt)
Integration by parts, which is int(u dv) = uv - int(v du), can work in reverse, even with a constant. The integral of 1 is indeed x.
I still say the problem is in the omission of constants. The indefinite integral of (1/x) is ln x + C. I can't stress the + C enough. Indefinite integrals do not equal each other because of these constants. The algebra fails:
int(1/x) dx = 1 + int(1/x) dx
ln x + C1 = 1 + ln x + C2
Because C1 and C2 can be anything, the problem is irrelevant. In fact, solving any indefinite integral in this problem yields an additional constant. The initial (1/x), for example, can have another constant.
Edited on July 8, 2004, 4:34 pm
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Posted by Eric
on 2004-07-08 16:31:09 |
re: Solution
