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I have faced a confusing maths problem that goes :

$f : [-1/2, 3/2] \to\mathbb R$ is a continuous function

Prove that :

$${\int_{-1/2}^{3/2} x f(3x^2 -2x^3) \,dx =2 \int_0^1 xf(3x^2 - 2x^3) dx}$$

I thought about using the formula : $${\int_a^b\ f(x) dx = \frac{a+b}2\int_a^b f(x)dx }$$

and the formula : $${\int_a^b\ f(x) dx =\int_a^b f(a+ b -x)dx }$$

Nevertheless, they both seem not to help. Even trying the hints mentioned in the comments is fruitless.

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    $\begingroup$ Hint: Try plotting the function $3x^2-2x^3$ and then look for a suiting set of substitutions. $\endgroup$ Oct 17, 2023 at 16:48
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    $\begingroup$ Just a guess if you have $a(x) = [g(x)+g(-x)]/2$ and $b(x)=[g(x)-g(-x)]/2$, Then $g(x)= a(x)+b(x)$ and $\int_{-c}^c b(x) dx =0, \ \int_{-c}^c a(x) dx = 2\int_0^{c} a(x)dx$ combined with $x=p(u)$ might do it. The doubling of second integral and the partial symmetry in the first version is suggestive. $\endgroup$ Oct 17, 2023 at 17:49
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    $\begingroup$ I second @MaximilianJanisch comment. $\endgroup$
    – homosapien
    Oct 17, 2023 at 20:24
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    $\begingroup$ Title's support MathJax. $\endgroup$
    – Kurt G.
    Nov 11, 2023 at 12:38
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    $\begingroup$ @Salem: Ah OK. I think you need some more calculus level for this problem (However i don't discard to be wrong about it). Regards. $\endgroup$
    – Piquito
    Dec 10, 2023 at 14:32

1 Answer 1

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HINT.- An idea leading to a proof. Denote $(*)$ the proposed equality.

$(1)$The equality $(*)$ is true for all monomial $f(x)=a_nx^n$.

In fact $$\displaystyle a_n\int_{-1/2}^{3/2} x(3x^2-2x^3)^ndx=2a_n\int_0^1x^{1+2n}(3-2x)^ndx$$ we can verify this for all integer $n$. (For example, discarding WLOG the constant coefficient $a_n$, we have in both integrals $(*)$ the values $1,0.7,0.405623$ for exponents $0,1,5$ respectively). This probably can be proven by induction on $n$.

$(2)$ Because of linearity of integrals, it follows that $(*)$ is true for all polynomial $f(x)$.

$(3)$Now, for Weirstrass approximation theorem, every continuous real function $f(x)$ defined on the compact $[-1/2,3/2]$ is uniformly limit of polynomials $P_n(x)$ and each of them satisfy $(*)$.

Full details can be given by anyone who wishes to complete the proof.

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