Question

The value of the integral $\int \frac{(x^2-1)dx}{x^3\left(\sqrt{2x^4-2x^2+1}\right)}$ is:

• A. $2\sqrt{2-\frac{2}{x^2}+\frac{1}{x^4}}+c$
• B. $2\sqrt{2+\frac{2}{x^2}+\frac{1}{x^4}}+c$
• C. $\frac{1}{2}\sqrt{2-\frac{2}{x^2}+\frac{1}{x^4}}+c$
• D. None of these

Answer 1

The correct option is C $\frac{1}{2}\sqrt{2-\frac{2}{x^2}+\frac{1}{x^4}}+c$

$I=\int \frac{(x^2-1)dx}{x^3\left(\sqrt{2x^4-2x^2+1}\right)}\Rightarrow I=\int \frac{x^2-1}{x^5\sqrt{2-\frac{2}{x^2}+\frac{1}{X^4}}}dx\Rightarrow I=\int \frac{\frac{1}{x^3}-\frac{1}{x^5}}{\sqrt{2-\frac{2}{x^2}+\frac{1}{X^4}}}dx\text{Put}2-\frac{2}{x^2}+\frac{1}{X^4}=t\Rightarrow (\frac{1}{x^3}-\frac{1}{x^5})dx=\frac{dt}{4}\Rightarrow I=\frac{1}{4}\int \frac{dt}{\sqrt{t}}\Rightarrow I=\frac{1}{2}\sqrt{t}+c\Rightarrow I=\frac{1}{2}\sqrt{2-\frac{2}{x^2}+\frac{1}{x^4}}+c$