Sets, Relations & Functions
Iterated Composition and Definite Integral
nta_pyq_2024_jan
Grade 11
Question:
Let $f:\mathbb{R}\to\mathbb{R}$ be a function defined $f(x)=\dfrac{x}{(1+x^4)^{1/4}}$ and $g(x)=f(f(f(f(x))))$, then $18\displaystyle\int_0^{\sqrt{2\sqrt{5}}}x^2g(x)\,dx$
Step-by-Step Solution
Key Concept: Compute $f\circ f(x)$: after two iterations, $f(f(x))=\frac{x}{(1+2x^4)^{1/4}}$. After four: $g(x)=\frac{x}{(1+4x^4)^{1/4}}$. Evaluate the integral by substitution $1+4x^4=t^4$.
$g(x)=\frac{x}{(1+4x^4)^{1/4}}$. $18\int_0^{\sqrt{2\sqrt5}}\frac{x^3}{(1+4x^4)^{1/4}}dx=\frac{3}{2}[t^3]_1^3=\frac{3}{2}\cdot26=39$.
Correct Answer: 4