`"The integral " int(dx)/(x^(2)(x^(4)+1)^(3//4))" equals"`

Options

A. `((x^(4)+1)/(x^(4)))^(1//4)+c`
B. `(x^(4)+1)^(1//4)+c`
C. `-(x^(4)+1)^(1//4)+c`
D. `-((x^(4)+1)/(x^(4)))^(1//4)+c`

Explanation

To solve the integral

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

we will follow these steps:

### Step 1: Rewrite the Integral
We start by rewriting the integral in a more manageable form. We can factor out \(x^4\) from the denominator:

\[ I = \int \frac{1}{x^2 (x^4 + 1)^{3/4}} \, dx = \int \frac{1}{x^2} \cdot \frac{1}{(x^4 + 1)^{3/4}} \, dx. \]

### Step 2: Factor Out \(x^4\)
Next, we factor out \(x^4\) from the term \((x^4 + 1)^{3/4}\):

\[ I = \int \frac{1}{x^2} \cdot \frac{1}{x^{3}} \cdot \frac{1}{(1 + \frac{1}{x^4})^{3/4}} \, dx. \]

This simplifies to:

\[ I = \int \frac{1}{x^5} \cdot \frac{1}{(1 + \frac{1}{x^4})^{3/4}} \, dx. \]

### Step 3: Substitution
Now, we will use the substitution:

\[ t = 1 + \frac{1}{x^4} \implies dt = -\frac{4}{x^5} \, dx \implies dx = -\frac{x^5}{4} \, dt. \]

From the substitution, we also have:

\[ \frac{1}{x^5} \, dx = -\frac{1}{4} \, dt. \]

### Step 4: Substitute Back into the Integral
Substituting \(t\) and \(dx\) back into the integral gives:

\[ I = \int -\frac{1}{4} t^{-3/4} \, dt. \]

### Step 5: Integrate
Now we can integrate:

\[ I = -\frac{1}{4} \cdot \left( \frac{t^{1/4}}{1/4} \right) + C = -t^{1/4} + C. \]

### Step 6: Substitute Back for \(t\)
Now substitute back for \(t\):

\[ I = -\left(1 + \frac{1}{x^4}\right)^{1/4} + C. \]

### Final Result
Thus, the final result for the integral is:

\[ I = -\left(1 + \frac{1}{x^4}\right)^{1/4} + C. \]

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