Animated Solution for Mathematics - Indefinite Integration: Evaluate the following $\int \frac{dx}{x^2(x^4+1{)}^{3/4}}$

Welcome, fellow traveler of the calculus path. Today, we are going to dismantle a seemingly terrifying integral:

That fractional power in the denominator looks like a nightmare, but in the world of JEE Advanced, intimidation is just a mask for a beautiful, hidden symmetry. Let's peel back the layers together.

The Algebraic Surgery

When you see a binomial like $(x^4+1)$ raised to a fractional power, your intuition should immediately scream "substitution." We need to manipulate the expression so that the term outside the bracket, the $x^2$, becomes the derivative of something inside.

The secret key here is to factor out the highest power of $x$ from the binomial, which is $x^4$:

Now, we apply the exponent $3/4$ to this entire product. By the laws of exponents, $(ab{)}^n=a^n{b}^n$, so we get:

The Perfect Substitution

Now, let's bring this back to our original integral. The $x^2$ in the denominator is now joined by the $x^3$ we just extracted, giving us $x^5$ in the denominator. Our integral transforms into:

This is ideal because the derivative of $1+x^{-4}$ is $-4x^{-5}$, which is exactly $-4/x^5$. We have successfully created the derivative right next to the function.

Let's define our substitution:

Differentiating both sides with respect to $x$, we get:

The Collapse

The rest is pure elegance. We substitute our pieces into the integral:

Pulling the constant $-1/4$ outside, we are left with:

This is a standard power rule integral. Adding one to the exponent $-3/4$ gives us $1/4$. Dividing by the new exponent $1/4$ is the same as multiplying by $4$:

Finally, we back-substitute $t=1+\frac{1}{x^4}$ to return to our original variable. The final result is:

The terror was just an illusion. By systematically breaking down the structure, we turned a complex expression into a simple power rule problem. Keep this "highest power factoring" technique in your toolkit; it will serve you well in many JEE battles to come.