Question

Let $f$ be a polynomial function such that for all real $x$ $f\left(x^2+1\right)=x^4+5 x^2+3$ then the premitive of $f( x )$ w.r.t. $x$, is

• A. $\frac{x^3}{3}+\frac{3 x^2}{2}-x+C$
• B. $\frac{x^3}{3}-\frac{3 x^2}{2}+x+C$
• C. $\frac{x^3}{3}-\frac{3 x^2}{2}-x+C$
• D. $\frac{x^3}{3}+\frac{3 x^2}{2}+x+C$

Answer 1

Answer: (A)

$f\left(x^2+1\right)=\left(x^2+1\right)^2+3 x^2+2=\left(x^2+1\right)^2+3\left(x^2+1\right)-1$
$\text { put } x ^2+1= t$
$ f ( t )= t ^2+3 t -1 \text { now integrate } $
$\therefore f ( x )= x ^2+3 x -1$