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The source for the inequality is
P.L. Chebyshev, On approximate expressions of some integrals in terms of others, taken within the same limits, Proc. Math. Soc. Kharkov 2, 93–98 (1882).

This article is online here, in Russian. The relevant equation is copied below, with the statement that $R_1=AB/12$ and $A,B$ being the largest of the derivatives of $du/dx$ and $dv/dx$, hence $R_1\geq 0$ for monotone $f,g$.
_{Someone who reads Russian will correct me if I misinterpreted the text.}

The source for the inequality is
P.L. Chebyshev, On approximate expressions of some integrals in terms of others, taken within the same limits, Proc. Math. Soc. Kharkov 2, 93–98 (1882).

This article is online here. The relevant text is copied below, which presumably states $R_1>0$, hopefully someone who reads Russian can provide a translation.
_{Google translate suggests that $R_1=AB/12$ with $A,B$ the maximum of the derivatives of $du/dx$, $dv/dx$, which as noted by Emil Jeřábek cannot be quite correct.}

The source for the inequality is
P.L. Chebyshev, On approximate expressions of some integrals in terms of others, taken within the same limits, Proc. Math. Soc. Kharkov 2, 93–98 (1882).

This article is online here, in Russian. The relevant equation is copied below, with the statement that $R_1=AB/12$ and $A,B$ being the largest of the derivatives of $du/dx$ and $dv/dx$, hence $R_1\geq 0$ for monotone $f,g$.
_{Someone who reads Russian will correct me if I misinterpreted the text.}

The source for the inequality is
P.L. Chebyshev, On approximate expressions of some integrals in terms of others, taken within the same limits, Proc. Math. Soc. Kharkov 2, 93–98 (1882).

This article is online here, in Russian. The relevant equation is copied below, with the statement that $R_1=AB/12$ and $A,B$ being the largest of the derivatives of $du/dx$ and $dv/dx$, hence $R_1\geq 0$ for monotone $f,g$.
_{Someone who reads Russian will correct me if I misinterpreted the text.}

The inequality is attributed to
P.L. Chebyshev, On approximate expressions of some integrals in terms of others, taken within the same limits, Proc. Math. Soc. Kharkov 2, 93–98 (1882).
_{more often cited with the title in French: "Sur les expressions approximatives des integrales definies par les autres prises entre les mêmes limites"}

This article is online here, in Russian. I am having some difficulty to locate the inequality. The equation should be the one below, with $R_1\geq 0$, hopefully someone who reads Russian can locate that statement.

The source for the inequality is
P.L. Chebyshev, On approximate expressions of some integrals in terms of others, taken within the same limits, Proc. Math. Soc. Kharkov 2, 93–98 (1882).

This article is online here, in Russian. The relevant equation is copied below, with the statement that $R_1=AB/12$ and $A,B$ being the largest of the derivatives of $du/dx$ and $dv/dx$, hence $R_1\geq 0$ for monotone $f,g$.
_{Someone who reads Russian will correct me if I misinterpreted the text.}