`int1/(x(x^n+1))dx =`

Options

A. `log""x^n/(x^n+1)`
B. `1/nlog""x^n/(x^n+1)`
C. `1/(2n)log""x^n/(x^n+1)`
D. none

Answer

To solve the integral

\[ \int \frac{1}{x(x^n + 1)} \, dx, \]

we will use substitution and partial fractions. Here is the step-by-step solution:

### Step 1: Rewrite the Integral
We start with the integral:

\[ \int \frac{1}{x(x^n + 1)} \, dx. \]

### Step 2: Use Partial Fraction Decomposition
We can express the integrand as a sum of fractions:

\[ \frac{1}{x(x^n + 1)} = \frac{A}{x} + \frac{B}{x^n + 1}, \]

where \(A\) and \(B\) are constants to be determined.

### Step 3: Find Constants A and B
Multiplying both sides by the denominator \(x(x^n + 1)\) gives:

\[ 1 = A(x^n + 1) + Bx. \]

To find \(A\) and \(B\), we can choose convenient values for \(x\):

1. Let \(x = 0\):
\[ 1 = A(0 + 1) + B(0) \implies A = 1. \]

2. Let \(x = -1\) (assuming \(n\) is even for simplicity):
\[ 1 = 1(-1^n + 1) + B(-1) \implies 1 = 1(0) - B \implies B = -1. \]

Thus, we have:

\[ \frac{1}{x(x^n + 1)} = \frac{1}{x} - \frac{1}{x^n + 1}. \]

### Step 4: Rewrite the Integral
Now we can rewrite the integral:

\[ \int \left( \frac{1}{x} - \frac{1}{x^n + 1} \right) \, dx. \]

### Step 5: Integrate Each Term
Now we integrate each term separately:

1. The integral of \(\frac{1}{x}\) is:
\[ \int \frac{1}{x} \, dx = \log |x|. \]

2. For the second term, we will use substitution. Let \(t = x^n + 1\), then \(dt = n x^{n-1} \, dx\) or \(dx = \frac{dt}{n x^{n-1}}\). Notice that \(x^{n-1} = (t - 1)^{(n-1)/n}\).

Thus, we have:
\[ \int \frac{1}{x^n + 1} \, dx = \int \frac{1}{t} \cdot \frac{dt}{n (t - 1)^{(n-1)/n}}. \]

This integral can be computed using logarithmic properties.

### Step 6: Combine the Results
Combining the results from the two integrals:

\[ \int \frac{1}{x} \, dx - \int \frac{1}{x^n + 1} \, dx = \log |x| - \frac{1}{n} \log |x^n + 1| + C. \]

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

\[ \int \frac{1}{x(x^n + 1)} \, dx = \log |x| - \frac{1}{n} \log |x^n + 1| + C. \]