Question

Trig Substitution, I dont understand anti derivative of (sqrt(x^2-4x+1)).

Answer 1

I dont understand how come the right side of the answer is -3/2 ln | (x-2)+sqrt(x^2-4+1) |+C

Answer 2

http://puu.sh/fIu6p/28469dcffb.png

Answer 3

This is the all the steps to the solution, but what happened to the sqrt(3) at the bottom?

Answer 4

It gets absorbed into C.
\[
\ln\left(\frac{a}{b}\right)=\ln(a)-\ln(b)
\]

Answer 5

How does that work?

Answer 6

is it because ln(b) is just some sort of constant that we don't care?

Answer 7

In other words, are all constants absorb into C?

Answer 8

\(\ln \sqrt{3}\) is a constant

Answer 9

and you must be knowing that \(C\) is an "arbitrary" constant

Answer 10

\(C\) is as much arbitrary as \(\ln\sqrt{3} + C\)

if you like, you may define a new constant : \(C' =\ln\sqrt{3} + C \) and use it

Answer 11

thats called lumping of constants

Answer 12

I understand this now, thanks a lot thomas & ganeshie

Answer 13

here is a fun 0=1 fake proof using this lumping trick

http://math.stackexchange.com/questions/806254/using-integration-by-parts-results-in-0-1