Let f be a polynomial function such that f(x^2+1) = x^4+5x^2+2, for all xR.

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:

Accepted Answer

Let $t=x^2+1$, so $x^2=t-1$:

$$f(t) = (t-1)^2+5(t-1)+2 = t^2-2t+1+5t-5+2 = t^2+3t-2$$

So $f(x)=x^2+3x-2$.

$$\int_0^3 f(x)\,dx = \left[\frac{x^3}{3}+\frac{3x^2}{2}-2x\right]_0^3 = 9+\frac{27}{2}-6 = 3+\frac{27}{2} = \frac{33}{2}$$

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