Step-by-step explanation:
Question
[tex]\displaystyle \bullet\ \quad \sf \red{\int \dfrac{1}{x\sqrt{x^n-1}}\ dx}∙ [/tex]
Solution
[tex]\displaystyle \sf \red{\int \dfrac{1}{x\sqrt{x^n-1}}\ dx}[/tex]
[tex]\displaystyle\int \sf \dfrac{x^{n-1}}{x^n\ \sqrt{x^n-1}}\ [/tex]
Lets use substitution method , where
[tex] \longmapsto\sf=\sqrt{x^n-1}u= x n −1[/tex]
[tex] \longmapsto \sf ^2=x^n-1u2 =x n −1[/tex]
[tex] \longmapsto\sf u^2+1=x^nu 2 [/tex]
[tex] \sf 2u\ \longmapsto \: du = nx^{n-1}[/tex]
[tex] \longmapsto\sf \dfrac{2u}{n}\ du=x^{n-1}[/tex]
[tex]\displaystyle \longrightarrow \quad \int \sf\dfrac{\frac{2u}{n}}{(u^2+1)\ u}[/tex]
[tex]\longrightarrow \quad \sf \dfrac{2}{n}\ \int \dfrac{1}{u^2+1}[/tex]
[tex]\displaystyle \longrightarrow \quad \sf \dfrac{2}{n}\ tan^{-1}(u)+c[/tex]
[tex]\displaystyle \quad \sf \pink{\longrightarrow \dfrac{2}{n}\ tan^{-1} \left( \sqrt{x^n-1} \right) +c}[/tex]
Hope It Helps!
