Let f: ℝ → ℝ be a function defined by f(x)=x/(1+x⁴)^(1 / 4), and g(x)=f(f(f(f(x)))). Then, 18 ∫₀^(√(2 √5)) x² g(x) d x is equal to

Solution

Maths sir · 4 steps

Compose f with itself and look at the pattern: each layer adds one more x⁴ under the fourth root.

1. Find the pattern. If h(x) = x/(1+kx⁴)^(1/4), then h⁴ = x⁴/(1+kx⁴), so 1 + h⁴ = (1+(k+1)x⁴)/(1+kx⁴). That gives f(h(x)) = x/(1+(k+1)x⁴)^(1/4), so each extra f raises k by 1.
2. Four layers. Start from f with k = 1 and apply f three more times. This gives g(x) = x/(1+4x⁴)^(1/4), so the integrand is x² g(x) = x³/(1+4x⁴)^(1/4).
3. Substitute. Put u = 1 + 4x⁴, so du = 16x³ dx. At x = 0, u = 1. At x = √2√5, x⁴ = 4 × 5 = 20, so u = 81.

   ∫₀^(√2√5) x³/(1+4x⁴)^(1/4) dx = 1/16∫₁⁸¹ u^(-1/4) du = 1/12[u^(3/4)]₁⁸¹ = (27-1)/12 = 13/6

4. So. 18 × 13/6 = 39, which is option (c).

The idea to keep. When a function is composed with itself many times, work out two layers by hand and spot the pattern. Here f applied n times gives x/(1+nx⁴)^(1/4).

Answer from the NTA final answer key. Solution by PrepWiser.