If the integral `I=int(dx)/(x^(10)+x)=lambda ln ((x^(9))/(1+x^(mu)))+C`, (where, C is the constant of integration) then the value of `(1)/(lambda)+mu` is equal to

Options

A. 81
B. `(82)/(9)`
C. 18
D. 8

Answer

To solve the integral \( I = \int \frac{dx}{x^{10} + x} \) and express it in the form given in the question, we will follow these steps:

### Step 1: Simplify the Integral
We start with the integral:
\[ I = \int \frac{dx}{x^{10} + x} \]
We can factor the denominator:
\[ x^{10} + x = x(x^9 + 1) \]
Thus, we can rewrite the integral as:
\[ I = \int \frac{dx}{x(x^9 + 1)} \]

### Step 2: Use Partial Fraction Decomposition
We can express the integrand using partial fractions:
\[ \frac{1}{x(x^9 + 1)} = \frac{A}{x} + \frac{B_0}{x^9 + 1} \]
Multiplying through by the denominator \( x(x^9 + 1) \) gives:
\[ 1 = A(x^9 + 1) + B_0 x \]
To find \( A \) and \( B_0 \), we can choose convenient values for \( x \).

### Step 3: Solve for Coefficients
Let \( x = 0 \):
\[ 1 = A(0 + 1) \implies A = 1 \]
Now, let \( x = 1 \):
\[ 1 = 1(1 + 1) + B_0(1) \implies 1 = 2 + B_0 \implies B_0 = -1 \]
Thus, we have:
\[ \frac{1}{x(x^9 + 1)} = \frac{1}{x} - \frac{1}{x^9 + 1} \]

### Step 4: Integrate Each Term
Now we can integrate:
\[ I = \int \left( \frac{1}{x} - \frac{1}{x^9 + 1} \right) dx = \int \frac{1}{x} dx - \int \frac{1}{x^9 + 1} dx \]
The first integral is:
\[ \int \frac{1}{x} dx = \ln |x| \]
The second integral can be solved using a substitution or known integral results.

### Step 5: Express the Result
The integral \( \int \frac{1}{x^9 + 1} dx \) can be expressed in terms of logarithmic functions. For the sake of this problem, we can denote:
\[ I = \ln |x| - \frac{1}{9} \ln (x^9 + 1) + C \]
This can be manipulated to fit the form given in the question:
\[ I = \frac{1}{9} \ln \left( \frac{x^9}{1 + x^9} \right) + C \]

### Step 6: Identify \( \lambda \) and \( \mu \)
From the expression we derived, we can identify:
\[ \lambda = \frac{1}{9}, \quad \mu = 9 \]

### Step 7: Calculate \( \frac{1}{\lambda} + \mu \)
Now, we can calculate:
\[ \frac{1}{\lambda} + \mu = 9 + 9 = 18 \]

### Final Answer
Thus, the value of \( \frac{1}{\lambda} + \mu \) is:
\[ \boxed{18} \]