Let `f(x)=(x)/((1+x^(n))^(1//n))` for `n ge 2` and `g(x)=underset("n times")underbrace(fofo ..."of"(x))`, then `int x^(n-2)g(x)dx` equals to

Options

A. `(1)/(n(n-1))(1+nx^(n))^(1-1/n)+C`
B. `(1)/(n-1)(1+nx^(n))^(1-1/n)+C`
C. `(1)/(n(n+1))(1+nx^(n))^(1+1/n)+C`
D. `(1)/(n+1)(1+nx^(n))^(1+1/n)+C`

Answer

To solve the problem, we need to find the integral \( \int x^{n-2} g(x) \, dx \), where \( g(x) = f(f(...f(x)...)) \) (n times) and \( f(x) = \frac{x}{(1 + x^n)^{1/n}} \).

### Step-by-Step Solution:

1. **Define the Function \( f(x) \)**:
\[ f(x) = \frac{x}{(1 + x^n)^{1/n}} \]

2. **Calculate \( f(f(x)) \)**:
To find \( g(x) \), we need to compute \( f(f(x)) \):
\[ f(f(x)) = f\left(\frac{x}{(1 + x^n)^{1/n}}\right) \]
Substitute \( f(x) \) into itself:
\[ f(f(x)) = \frac{\frac{x}{(1 + x^n)^{1/n}}}{\left(1 + \left(\frac{x}{(1 + x^n)^{1/n}}\right)^n\right)^{1/n}} \]
Simplifying the denominator:
\[ = \frac{x}{(1 + x^n)^{1/n}} \cdot \left(1 + \frac{x^n}{(1 + x^n)}\right)^{1/n} = \frac{x}{(1 + x^n)^{1/n}} \cdot \left(\frac{1 + 2x^n}{1 + x^n}\right)^{1/n} \]
Continuing this process, we find:
\[ g(x) = f(f(...f(x)...)) = \frac{x}{(1 + nx^n)^{1/n}} \]

3. **Set Up the Integral**:
Now we need to evaluate:
\[ I = \int x^{n-2} g(x) \, dx = \int x^{n-2} \frac{x}{(1 + nx^n)^{1/n}} \, dx = \int \frac{x^{n-1}}{(1 + nx^n)^{1/n}} \, dx \]

4. **Substitution**:
Let \( t = x^n \), then \( dt = n x^{n-1} \, dx \) or \( dx = \frac{dt}{n x^{n-1}} = \frac{dt}{n t^{(n-1)/n}} \).
Substitute into the integral:
\[ I = \int \frac{t^{(n-1)/n}}{(1 + nt)^{1/n}} \cdot \frac{dt}{n t^{(n-1)/n}} = \frac{1}{n} \int \frac{1}{(1 + nt)^{1/n}} \, dt \]

5. **Evaluate the Integral**:
The integral \( \int \frac{1}{(1 + nt)^{1/n}} \, dt \) can be evaluated using the formula for integrals of the form \( \int (a + bx)^{-k} \, dx \):
\[ = \frac{(1 + nt)^{1/n - 1}}{(1/n - 1)n} + C \]
Thus,
\[ I = \frac{1}{n} \cdot \frac{(1 + nt)^{1/n - 1}}{(1/n - 1)n} + C \]

6. **Back Substitute**:
Replace \( t \) back with \( x^n \):
\[ I = \frac{1}{n^2} (1 + nx^n)^{1/n - 1} + C \]

### Final Answer:
\[ I = \frac{1}{n^2} (1 + nx^n)^{1/n - 1} + C \]