Antiderivatives: Calculus II Question?

Question

How would you begin anti-differentiating this problem?
dx / ( sqrt(x) * ( 1 - 2*sqrt(x) )

Thank you!

Answer 1

i would use substitution:
u= 1-2*sqrt(x)
du = 2(1/2) x^{-1/2} dx = dx/sqrt(x)

then
integral of dx/( sqrt(x) * ( 1 - 2*sqrt(x) )
= integral du/u = ln(u)+C = ln( 1-2sqrt(x) ) + C .

Answer 2

Try saying 'integrating' instead of 'anti-differentiating'... It sounds more like what a mathematician would say ☺

Next, re-write it as follows:
dx/(√x - 2x) and integrate to get
-ln(2√x - 1) +C

Now do yourself a favor and take the derivative of that to see how is 'unwinds' to the original function. It'll help you 'see' the general form of the answer to that kind of problem next time you run into it.

99% of being hot at integration is practice and good intuition (or dumb luck ☺)


Doug