Question

Let f (x) = \(\frac{x}{ (1+x ^n )^{\frac{1}{n} }}\), x∈R-{-1}, n∈N. n>2. 
If fⁿ (x) = (fofof____upto n times) (x), then \(lim ∫^1_ {n→∞ 0} x^{n-2}\)( f ⁿ(x))dx is equal to ___.

Answer 1

Correct Answer: 0

Given the functional equation:

\[ f(x) + f(\pi - x) = \pi^2. \]

Consider the given integral:

\[ I = \int_0^{\pi} f(x) \sin x \,dx. \]

Using the substitution \( t = \pi - x \), we get:

\[ I = \int_0^{\pi} f(\pi - x) \sin (\pi - x) \,dx. \]

Since \( \sin (\pi - x) = \sin x \), we obtain:

\[ I = \int_0^{\pi} f(\pi - x) \sin x \,dx. \]

Adding both integrals:

\[ 2I = \int_0^{\pi} (f(x) + f(\pi - x)) \sin x \,dx. \]

Substituting \( f(x) + f(\pi - x) = \pi^2 \):

\[ 2I = \pi^2 \int_0^{\pi} \sin x \,dx. \]

Since \( \int_0^{\pi} \sin x \,dx = 2 \), we get:

\[ 2I = \pi^2 \times 0 = 0. \]

Thus, \( I = 0 \).

Final Answer: \( \mathbf{0} \).