`int (dx)/(x^(2)(x^(4)+1)^(3/4))is`

Options

A. `-(1+(1)/(x^(4)))^(1/4)+c`
B. `(1)/(4)log|(x^(2))/(x^(4)+1)|+c`
C. `(1)/(2)log|x^(4)+1|+c`
D. `(1+(1)/(x^(4)))^((3)/(4))+c`

Answer

To solve the integral

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

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 term \((x^4 + 1)^{3/4}\):

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

### Step 2: Simplify the Expression
Now, we can simplify the expression:

\[ = \int \frac{dx}{x^2 \cdot x^{3} \cdot \left(1 + \frac{1}{x^4}\right)^{3/4}} = \int \frac{dx}{x^{5} \left(1 + \frac{1}{x^4}\right)^{3/4}}. \]

### Step 3: Substitute
Let \(t = 1 + \frac{1}{x^4}\). Then, we differentiate to find \(dt\):

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

### Step 4: Substitute Back into the Integral
Now we substitute \(dx\) and \(t\) into the integral:

\[ \int \frac{-\frac{x^5}{4} dt}{x^5 \cdot t^{3/4}} = -\frac{1}{4} \int \frac{dt}{t^{3/4}}. \]

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

\[ -\frac{1}{4} \int t^{-3/4} dt = -\frac{1}{4} \cdot \left(\frac{t^{1/4}}{1/4}\right) + C = -t^{1/4} + C. \]

### Step 6: Substitute Back for \(t\)
Finally, we substitute back for \(t\):

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

### Final Answer
Thus, the final answer is:

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

---