Question

\bf \orange{\dagger \quad Hola\ Guys\ !}" alt="\bf \orange{\dagger \quad Hola\ Guys\ !}" align="absmiddle" class="latex-formula">
$\ \textless \ br /\ \textgreater \ \bullet\ \quad \displaystyle \rm \int \dfrac{1}{x\sqrt{x^n-1}}$
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Answer 1

Hola mate..........

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Answer 2

Step-by-step explanation:

Question

$\displaystyle \bullet\ \quad \sf \red{\int \dfrac{1}{x\sqrt{x^n-1}}\ dx}∙$

Solution

$\displaystyle \sf \red{\int \dfrac{1}{x\sqrt{x^n-1}}\ dx}$

$\displaystyle\int \sf \dfrac{x^{n-1}}{x^n\ \sqrt{x^n-1}}$

Lets use substitution method , where

$\longmapsto\sf=\sqrt{x^n-1}u= x n −1$

$\longmapsto \sf ^2=x^n-1u2 =x n −1$

$\longmapsto\sf u^2+1=x^nu 2$

$\sf 2u\ \longmapsto \: du = nx^{n-1}$

$\longmapsto\sf \dfrac{2u}{n}\ du=x^{n-1}$

$\displaystyle \longrightarrow \quad \int \sf\dfrac{\frac{2u}{n}}{(u^2+1)\ u}$

$\longrightarrow \quad \sf \dfrac{2}{n}\ \int \dfrac{1}{u^2+1}$

$\displaystyle \longrightarrow \quad \sf \dfrac{2}{n}\ tan^{-1}(u)+c$

$\displaystyle \quad \sf \pink{\longrightarrow \dfrac{2}{n}\ tan^{-1} \left( \sqrt{x^n-1} \right) +c}$

Hope It Helps!