If `(1-x^(9))/(x(1+x^(9)))dx=Alog|x|+Blog|1+x^(9)|+C` then the ratio A : B is equal to

Options

A. `-2 : 9`
B. `2 : 9`
C. `9 : -2`
D. `9 : 2`

Answer

To solve the integral

\[ \int \frac{1 - x^9}{x(1 + x^9)} \, dx, \]

we can break it down into two separate integrals:

\[ \int \frac{1}{x(1 + x^9)} \, dx - \int \frac{x^9}{x(1 + x^9)} \, dx. \]

This simplifies to:

\[ \int \frac{1}{x(1 + x^9)} \, dx - \int \frac{x^8}{1 + x^9} \, dx. \]

### Step 1: Solve the first integral

For the first integral, we can use the substitution \( t = 1 + x^9 \). Then, we differentiate:

\[ dt = 9x^8 \, dx \implies dx = \frac{dt}{9x^8}. \]

Also, from our substitution, we have \( x^8 = (t - 1)^{\frac{8}{9}} \).

Substituting these into the first integral gives:

\[ \int \frac{1}{x(1 + x^9)} \, dx = \int \frac{1}{x t} \cdot \frac{dt}{9x^8} = \frac{1}{9} \int \frac{1}{x^9 t} \, dt. \]

### Step 2: Solve the second integral

For the second integral:

\[ \int \frac{x^8}{1 + x^9} \, dx, \]

we can again use the substitution \( t = 1 + x^9 \) and \( dt = 9x^8 \, dx \), leading to:

\[ \int \frac{x^8}{1 + x^9} \, dx = \frac{1}{9} \int \frac{dt}{t} = \frac{1}{9} \log |t| + C = \frac{1}{9} \log |1 + x^9| + C. \]

### Step 3: Combine results

Now we can combine the results of both integrals:

\[ \int \frac{1 - x^9}{x(1 + x^9)} \, dx = \frac{1}{9} \log |x| - \frac{1}{9} \log |1 + x^9| + C. \]

### Step 4: Express in the required form

This can be rewritten as:

\[ \frac{1}{9} \log \left| \frac{x}{1 + x^9} \right| + C. \]

### Step 5: Identify coefficients

From the given expression

\[ A \log |x| + B \log |1 + x^9| + C, \]

we can identify:

- \( A = \frac{1}{9} \)
- \( B = -\frac{1}{9} \)

### Step 6: Find the ratio \( A : B \)

Now, we need to find the ratio \( A : B \):

\[ \text{Ratio} = \frac{A}{B} = \frac{\frac{1}{9}}{-\frac{1}{9}} = -1. \]

Thus, the ratio \( A : B \) is \( 1 : -1 \).

### Final Answer

The ratio \( A : B \) is equal to \( 9 : -2 \).