Proof that $F(x)$ is an even function $f(x)$ is in $({-\infty},{+\infty})$ Continuous even function, and $F(x)=\int_0^x {(x-2t) \times f(t)dt}$.

Question

$f(x)$ is in $({-\infty},{+\infty})$ Continuous even function, and $F(x)=\int_0^x {(x-2t) \times f(t)dt}$. Proof that $F(x)$ is an even function.

Answer 1

$$F(-x) = \int_{0}^{-x}(-x-2t)\ f(t)\ dt$$

$t=-s$ : $$ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =-\int_{0}^{x}(-x+2s)\ f(-s)\ ds$$

$$\ \ \ \ \ \ \\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =\int_{0}^{x}(x-2s)\ f(s)\ ds$$

$$=F(x)$$

Therefore, $F$ is even.