Definite Integration
Functional Equation — Finding ∫f(x)dx
nta_pyq_2026_jan
Grade 12
Question:
Let $f$ be a polynomial function such that $f(x^2+1)=x^4+5x^2+2$, for all $x\in\mathbb{R}$. Then $\displaystyle\int_0^3 f(x)\,dx$ is equal to
\dfrac{5}{3}
\dfrac{41}{3}
\dfrac{27}{2}
\dfrac{33}{2}
Step-by-Step Solution
Key Concept: Let $u=x^2+1$ so $x^2=u-1$. Then $f(u)=(u-1)^2+5(u-1)+2=u^2-2u+1+5u-5+2=u^2+3u-2$.
$f(x)=x^2+3x-2$. $\int_0^3 f(x)\,dx=33/2$.
Correct Answer: 4