Let f (x) = $\frac{x}{ (1+x ^n )^{\frac{1}{n} }}$ , x∈R-{-1}, n∈N. n2. If f n (x) = (fofof____upto n times) (x), then $lim ∫^1_ {n→∞ 0} x^{n-2}$ ( f n (x))dx is equal to ___.

Question

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

cdquestions Admin · Accepted Answer

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} $.

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 \(lim ∫^1_ {n→∞ 0} x^{n-2}\)( fⁿ(x))dx is equal to ___.

Correct Answer: 0

Solution and Explanation

Top Questions on types of differential equations

Questions Asked in JEE Main exam

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 ___.

Correct Answer: 0

Solution and Explanation

Top Questions on types of differential equations

Questions Asked in JEE Main exam

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 \(lim ∫^1_ {n→∞ 0} x^{n-2}\)( fⁿ(x))dx is equal to ___.