Using integration by parts,
∫1/x dx = ∫1/x.1 dx
= ∫1/x. x-∫(-1/x^2)x dx
= ∫1+∫1/x dx.
subtracting ∫1/x dx from both sideswe conclude that 0=1. what is wrong with this logic?

Integrating by parts with

gives

But the two integrals don't cancel, because there are infinitely many antiderivatives of that differ by a constant.

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if she was to only spend a dollar she can get 1lb

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