Integration breaking down? Or am I being thick?

Question

So 1/5 / P is the same as 1/5P however they integrate to 1/5lnP and 1/5ln5P respectively, what's with this differnence?

Answer 1

Antiderivative of (1/5)/p dp is (1/5) ln p + C

 

Antiderivative of 1/(5p)dp can be done by u-sub: let u = 5p so du = 5 dp so dp = (1/5)du;

therefore ∫ (1/u)(1/5)du = (1/5) ln u + K where K is some arbitrary constant = (1/5)ln(5p) + K =

(1/5)[ln 5 + ln p] + K = (1/5) ln p  +  [(1/5)ln 5 + K]

 

C = [(1/5)ln 5 + K]...different forms of the arbitrary constant! 

Problem resolved...antidifferentiation is still believable.