Mathematics - Indefinite Integration Question with Solution | TestHub

Let $I=\int \frac{\left(x^2-1\right)dx}{x^3\sqrt{2x^4-2x^2+1}}$, On dividing numerater and denominator by $x^5$, we get $$=\int \frac{\left(\frac{1}{x^3}-\frac{1}{x^5}\right)dx}{\sqrt{2-\frac{2}{x^2}+\frac{1}{x^4}}}$$ put $2-\frac{2}{x^2}+\frac{1}{x^4}=t\Rightarrow \left(\frac{4}{x^3}-\frac{4}{x^5}\right)dx=dt$ $$\therefore I=\frac{1}{4}\int \frac{dt}{\sqrt{t}}=\frac{1}{4}\cdot \frac{t^{1/2}}{1/2}=\frac{1}{2}\sqrt{t}+c=\frac{1}{2}\sqrt{2-\frac{2}{x^2}+\frac{1}{x^4}}+C$$