Animated Solution for Mathematics - Indefinite Integration: $\int \frac{x^2-1}{x^3\sqrt{2x^4-2x^2+1}}dx=$

The integral we are tasked to solve is:

At first glance, this expression appears daunting. However, in JEE Advanced mathematics, the most intimidating problems often hide elegant structural symmetries. We will avoid brute force and instead employ algebraic manipulation to reveal the underlying pattern.

The key to this problem lies inside the square root: $2x^4-2x^2+1$. The standard strategy for such polynomials is to factor out the highest power of $x$ to force the derivative of the inner terms to appear in the numerator.

We factor out $x^4$ from the radical:

Pulling $x^4$ out of the square root as $x^2$, the expression becomes $x^2\sqrt{2-\frac{2}{x^2}+\frac{1}{x^4}}$. Substituting this back into the original denominator $x^3\sqrt{2x^4-2x^2+1}$, we obtain:

Now, we rewrite the integral $I$ by dividing the numerator $(x^2-1)$ by $x^5$:

Observe that the numerator is now proportional to the derivative of the expression inside the square root. Let us perform the substitution $t=2-2x^{-2}+x^{-4}$.

Differentiating with respect to $x$:

This implies that $\left(\frac{1}{x^3}-\frac{1}{x^5}\right)dx=\frac{dt}{4}$.

We have successfully transformed the integral into a simple power rule problem:

Integrating $t^{-1/2}$ yields $2\sqrt{t}$. Therefore:

Substituting $t=2-\frac{2}{x^2}+\frac{1}{x^4}$ back into the equation, we get:

Simplifying the denominator, we arrive at the final result:

Remember, in JEE, it is never about brute force; it is about finding the hidden structure.