The value of `int(dx)/(x(x^(n)+1))` is equal to

Options

A. `(1)/(n)log((x^(n))/(x^(n)+1))+C`
B. `log((x^(n)+1)/(x^(n)))+C`
C. `(1)/(n)log((x^(n)+1)/(x^(n)))+C`
D. `log((x^(n))/(x^(n)+1))+C`

Explanation

To solve the integral \( \int \frac{dx}{x(x^n + 1)} \), we will follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ \int \frac{dx}{x(x^n + 1)} \]
To simplify this, we can multiply and divide by \( x^{n-1} \):
\[ \int \frac{x^{n-1} \, dx}{x^n(x^n + 1)} = \int \frac{dx}{x^n + 1} \]

### Step 2: Substitute Variables
Let \( t = x^n \). Then, the differential \( dt = n x^{n-1} \, dx \) or \( dx = \frac{dt}{n x^{n-1}} \). Since \( x = t^{1/n} \), we have \( x^{n-1} = t^{(n-1)/n} \). Thus, we can rewrite \( dx \):
\[ dx = \frac{dt}{n t^{(n-1)/n}} \]
Substituting this into the integral gives:
\[ \int \frac{1}{t + 1} \cdot \frac{dt}{n t^{(n-1)/n}} = \frac{1}{n} \int \frac{dt}{t + 1} \]

### Step 3: Integrate
Now we can integrate:
\[ \frac{1}{n} \int \frac{dt}{t + 1} = \frac{1}{n} \log |t + 1| + C \]

### Step 4: Substitute Back
Recall that \( t = x^n \). Therefore, substituting back gives:
\[ \frac{1}{n} \log |x^n + 1| + C \]

### Final Answer
Thus, the value of the integral \( \int \frac{dx}{x(x^n + 1)} \) is:
\[ \frac{1}{n} \log |x^n + 1| + C \]