If the primitive of the function `f(x)=(x^(2009))/((1+x^(2))^(1006))` w.r.t. x is equal to `1/n ((x^(2))/(1+x^(2)))^(m)+C`, then `n/m` is equal to .......

Answer

To solve the problem, we need to find the integral of the function \( f(x) = \frac{x^{2009}}{(1 + x^2)^{1006}} \) and express it in the form given in the question. Let's go through the steps one by one.

### Step 1: Set up the integral
We start with the integral:
\[ I = \int \frac{x^{2009}}{(1 + x^2)^{1006}} \, dx \]

### Step 2: Simplify the integrand
We can factor out \( x^2 \) from the denominator:
\[ I = \int \frac{x^{2009}}{(1 + x^2)^{1006}} \, dx = \int \frac{x^{2009}}{x^2(1 + \frac{1}{x^2})^{1006}} \, dx = \int \frac{x^{2007}}{(1 + \frac{1}{x^2})^{1006}} \, dx \]

### Step 3: Substitute
Let \( t = 1 + \frac{1}{x^2} \). Then, we differentiate both sides:
\[ dt = -\frac{2}{x^3} \, dx \quad \Rightarrow \quad dx = -\frac{x^3}{2} \, dt \]
From our substitution, we have:
\[ x^2 = \frac{1}{t - 1} \quad \Rightarrow \quad x^3 = \left(\frac{1}{t - 1}\right)^{3/2} \]

### Step 4: Rewrite the integral in terms of \( t \)
Substituting \( x^3 \) and \( dx \) into the integral, we have:
\[ I = \int \frac{\left(\frac{1}{t - 1}\right)^{2007/2}}{t^{1006}} \left(-\frac{\left(\frac{1}{t - 1}\right)^{3/2}}{2}\right) dt \]
This simplifies to:
\[ I = -\frac{1}{2} \int \frac{1}{(t - 1)^{1005}} \cdot \frac{1}{t^{1006}} \, dt \]

### Step 5: Solve the integral
Now we can integrate:
\[ I = -\frac{1}{2} \cdot \int (t - 1)^{-1005} t^{-1006} \, dt \]
Using the formula for integration, we find:
\[ I = -\frac{1}{2} \cdot \left( \frac{(t - 1)^{-1004}}{-1004} \cdot \frac{1}{t^{1006}} \right) + C \]

### Step 6: Substitute back for \( t \)
Substituting back \( t = 1 + \frac{1}{x^2} \):
\[ I = -\frac{1}{2} \cdot \left( \frac{(1 + \frac{1}{x^2} - 1)^{-1004}}{-1004} \cdot \frac{1}{(1 + \frac{1}{x^2})^{1006}} \right) + C \]
This simplifies to:
\[ I = \frac{1}{2008} \cdot \frac{x^2}{(1 + x^2)^{1005}} + C \]

### Step 7: Compare with the given form
The expression we found can be compared to the form given in the question:
\[ \frac{1}{n} \left( \frac{x^2}{1 + x^2} \right)^{m} + C \]
From our result, we have:
- \( n = 2008 \)
- \( m = 1005 \)

### Step 8: Calculate \( \frac{n}{m} \)
Thus, we find:
\[ \frac{n}{m} = \frac{2008}{1005} = \frac{2}{1} = 2 \]

### Final Answer
The value of \( \frac{n}{m} \) is \( 2 \).