Prove-that-6-11-dx-x-2-x-3-2ln-3-2-2-2-3-

Question Number 17472 by Arnab Maiti last updated on 06/Jul/17

Prove that \int_{6}^{11} \frac{dx}{\sqrt{(x - 2) (x - 3)}} = 2 \ln \frac{3 + 2 \sqrt{2}}{2 + \sqrt{3}}

Answered by sma3l2996 last updated on 06/Jul/17

I = \int_{6}^{11} \frac{d x}{\sqrt{x^{2} - 5 x + 6}} = \int_{6}^{11} \frac{d x}{\sqrt{{(x - \frac{5}{2})}^{2} - \frac{1}{4}}} = 2 \int_{6}^{11} \frac{d x}{\sqrt{{(2 x - 5)}^{2} - 1}}

u = 2 x - 5 \Rightarrow d u = 2 d x

I = \int_{7}^{17} \frac{d u}{\sqrt{u^{2} - 1}}

u = c o s h t \Rightarrow d u = s i n h t d t = \sqrt{u^{2} - 1} d t \Leftrightarrow d t = \frac{d u}{\sqrt{u^{2} - 1}}

I = \int_{{c o s h}^{- 1} (7)}^{{c o s h}^{- 1} (17)} d t = {c o s h}^{- 1} (17) - {c o s h}^{- 1} (7)

= l n (\frac{17 + \sqrt{17^{2} - 1}}{7 + \sqrt{7^{2} - 1}}) = l n (\frac{17 + 12 \sqrt{2}}{7 + 4 \sqrt{3}})

Commented by Arnab Maiti last updated on 07/Jul/17

I think to prove the answer we need to put

(x - 3) = z^{2}