Question

$\bf \orange{\dagger \quad Hola\ Guys\ !}$
$\bullet\ \quad \displaystyle \rm \int \dfrac{1}{x\sqrt{x^n-1}}\ dx$

Answer 1

Question :

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

Solution :

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

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

Lets use substitution method , where

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

⇒ $\sf u^2 = x^n-1$

⇒ $\sf u^2+1=x^n$

⇒ $\sf 2u\ du = nx^{n-1}\ dx$

⇒ $\sf \dfrac{2u}{n}\ du=x^{n-1}\ dx$

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

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

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

• $\sf u = \sqrt{x^n-1}$

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

Answer 2

Question!

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

Solution!

• Answer refer to the attachment