If-f-x-lim-x-x-n-x-n-x-n-x-n-x-gt-1-then-xf-x-ln-x-1-x-2-1-x-2-dx-

Question Number 128540 by bramlexs22 last updated on 08/Jan/21

If f (x) = \lim_{x \to \infty} \frac{x^{n} - x^{- n}}{x^{n} + x^{- n}}, x > 1

then \int \frac{xf (x) \ln (x + \sqrt{1 + x^{2}})}{\sqrt{1 + x^{2}}} dx = ?

Commented by liberty last updated on 08/Jan/21

f (x) = \lim_{x \to \infty} \frac{x^{2 n} - 1}{x^{2 n} + 1} = 1

then \int \frac{x . 1 \ln (x + \sqrt{1 + x^{2}})}{\sqrt{x^{2} + 1}} dx

let \sqrt{1 + x^{2}} = u \Rightarrow \frac{x dx}{\sqrt{1 + x^{2}}} du and x = \sqrt{u^{2} - 1}

\int \ln (u + \sqrt{u^{2} - 1}) du; by parts

= u \ln (u + \sqrt{u^{2} - 1}) - \int \frac{u (1 + \frac{u}{\sqrt{u^{2} - 1}})}{u + \sqrt{u^{2} - 1}} du

= u \ln (u + \sqrt{u^{2} - 1}) - \int \frac{u (u + \sqrt{u^{2} - 1}) (u - \sqrt{u^{2} - 1})}{- 1 \sqrt{u^{2} - 1}} du

= u \ln (u + \sqrt{u^{2} - 1}) + \int \frac{u}{\sqrt{u^{2} - 1}} du

= u \ln (u + \sqrt{u^{2} - 1}) + \sqrt{u^{2} - 1} + c

= \sqrt{1 + x^{2}} \ln (x + \sqrt{x^{2} + 1}) + x + c

Answered by mathmax by abdo last updated on 10/Jan/21

\frac{x^{n} - x^{- n}}{x^{n} + x^{- n}} = \frac{x^{2 n} - 1}{x^{2 n} + 1} we have x > 1 \Rightarrow \lim_{x \to + \infty} \frac{x^{n} - x^{- n}}{x^{n} + x^{- n}} = 1 \Rightarrow

I = \int \frac{xf (x) \ln (x + \sqrt{1 + x^{2}})}{\sqrt{1 + x^{2}}} dx = \int \frac{xln (x + \sqrt{1 + x^{2}})}{\sqrt{1 + x^{2}}} dx we do the ch .

x = sht \Rightarrow I = \int \frac{shtln (sht + cht)}{cht} cht = \int sh (t) \ln (e^{t}) dt

= \int t \times sh (t) dt = tch (t) - \int ch (t) dt = tch (t) - sht C

= argsh (x) \sqrt{1 + x^{2}} - x + C

= \sqrt{1 + x^{2}} \ln (x + \sqrt{1 + x^{2}}) - x + C