How can we find the integral of 1/x?

How can we find the integral of 1/x?

If y(x) is a definite integral of 1/x then we have


dy/dx = 1/x


Therefore


dx/dy = x


From which we we know


x = ce^y


(Why? It's the same as saying the solution to f’(y)=f is f(y)=ce^y)


Taking log of both sides


lnx = lnc + y


Which shows that the integral is lnx plus an arbitrary constant, y = lnx + k


 


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