The integral ∫ dx / (x²(x⁴ + 1)^(3/4)) equals (a) -(x⁴ + 1)^(1/4) + c (b) -((x⁴ + 1)/x⁴)^(1/4) + c (

Q: Why do we factor out $x^4$ instead of just $x$? It feels like I am making it more complicated.

That is a brilliant question! If you factor out just $x$, you get $x(x^3 + 1/x)$, which does not help you create the derivative of the inner term. We factor out $x^4$ specifically because the derivative of $x^{-4}$ is $-4x^{-5}$. By factoring out the highest power, we are essentially 'engineering' the derivative to appear right in front of us. It is not making it complicated; it is making it solvable.

Q: How do I know when to use this specific substitution method?

Look for the 'signature' of this problem: a polynomial raised to a fractional power in the denominator, multiplied by a power of $x$. Whenever you see $x^n(ax^m + b)^p$, this 'factor out the highest power' strategy is your go-to move. It is a classic JEE pattern—once you recognize the pattern, the problem stops being a puzzle and starts being a procedure.

Q: Wait, why did the $x^5$ move to the numerator? Is that allowed?

Absolutely! Remember that $\frac{1}{x^5}$ is mathematically identical to $x^{-5}$. We move it to the numerator because it makes the substitution $t = 1 + x^{-4}$ much easier to visualize. When you see $x^{-5} dx$ in the numerator and $x^{-4}$ inside the bracket, you know immediately that the derivative of the bracket is sitting right there waiting for you. It is all about making the calculus 'see' the substitution clearly.

Question

The integral $\int \frac{dx}{x^2(x^4 + 1)^{3/4}}$ equals

Options:

(A) $-(x^4 + 1)^{1/4} + c$

(B) $-\left( \frac{x^4 + 1}{x^4} \right)^{1/4} + c$

(C) $\left( \frac{x^4 + 1}{x^4} \right)^{1/4} + c$

(D) $(x^4 + 1)^{1/4} + c$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

Welcome, future engineers. Today, we are going to dismantle a problem that often intimidates students at first glance: the integral

$$I = \int \frac{dx}{x^2(x^4 + 1)^{3/4}}$$

It looks messy, does it not? We have a polynomial trapped inside a fractional power, sitting in the denominator, multiplied by an $x^2$.

In the world of JEE Advanced, these problems are rarely about brute force; they are about recognizing the **hidden derivative**.

### The Algebraic Surgery

The first step in our journey is what I like to call 'Algebraic Surgery.' We need to liberate the term inside the bracket, $(x^4 + 1)^{3/4}$.

The golden key to this problem is to factor out the highest power of $x$ from inside that bracket. Let us rewrite the term:

$$(x^4 + 1)^{3/4} = [x^4(1 + x^{-4})]^{3/4}$$

Now, we distribute the power of $3/4$ to both factors. This gives us $(x^4)^{3/4} \cdot (1 + x^{-4})^{3/4}$.

The term $(x^4)^{3/4}$ simplifies beautifully to $x^3$. When we multiply this by the $x^2$ already in the denominator, we get $x^5$.

The denominator is now $x^5(1 + x^{-4})^{3/4}$. The chaos is beginning to organize itself.

### The Magic of Substitution

Now that we have

$$I = \int \frac{dx}{x^5(1 + x^{-4})^{3/4}}$$

we move the $x^5$ to the numerator as $x^{-5}$. Our integral now looks like:

$$I = \int \frac{x^{-5} dx}{(1 + x^{-4})^{3/4}}$$

Do you see it now? If we differentiate the term inside the bracket, $1 + x^{-4}$, we get $-4x^{-5}$.

We have an $x^{-5}$ sitting right there in the numerator! This is the **'Aha!' moment**.

Let us define our substitution: $t = 1 + x^{-4}$. Differentiating both sides with respect to $x$, we get:

$$\frac{dt}{dx} = -4x^{-5} \Rightarrow x^{-5} dx = -\frac{1}{4} dt$$

### The Final Victory

We are in the home stretch. Substituting these values back into our integral, the entire expression transforms into:

$$I = \int \frac{-\frac{1}{4} dt}{t^{3/4}}$$

This is a simple power rule problem. We pull out the constant $-\frac{1}{4}$ and integrate $t^{-3/4}$ to get:

$$I = -\frac{1}{4} \cdot \frac{t^{1/4}}{1/4} + c = -t^{1/4} + c$$

Finally, we substitute back $t = 1 + x^{-4}$. We get $-(1 + x^{-4})^{1/4} + c$.

To match standard forms, we rewrite $1 + x^{-4}$ as $\frac{x^4 + 1}{x^4}$. Thus, our final answer is:

$$\mathbf{-\left(\frac{x^4 + 1}{x^4}\right)^{1/4} + c}$$

What started as a terrifying, complex expression was actually a beautifully orchestrated dance of algebra and calculus. Keep practicing this pattern recognition—it is the hallmark of a true problem solver.

Animated Solution for Mathematics - Indefinite Integration: The integral $\int \frac{d x}{x ^{2} ( x ^{4} + 1 ) ^{3/4}}$ equals

Select Answer:

Visualized Solution

The Integral Setup

Strategy: Factoring out $x^{4}$

Algebraic Manipulation

Simplifying the Denominator

Rewriting with Negative Exponents

Defining the Substitution

Differentiating the Substitution

Substituting into the Integral

Integrating with Power Rule

Back-Substitution and Final Form

The Sigma Insight: Integration by Substitution

Analyzing the Setup

The Algebraic Surgery

The Magic of Substitution

The Final Victory

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Animated Solution for Mathematics - Indefinite Integration: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The integral ∫x2(x4+1)3/4dx​ equals