# 两个行列式定积分不等式的证明

\begin{equation*}\begin{vmatrix}
\int_{0}^{1}x^2\,\mathrm{d}x&\int_{0}^{1}x^3\,\mathrm{d}x &\int_{0}^{1}x^4\,\mathrm{d}x&\int_{0}^{1}xe^x\,\mathrm{d}x\\[4mm]
\int_{0}^{1}x^3\,\mathrm{d}x&\int_{0}^{1}x^4\,\mathrm{d}x &\int_{0}^{1}x^5\,\mathrm{d}x&\int_{0}^{1}x^2e^x\,\mathrm{d}x\\[4mm]
\int_{0}^{1}x^4\,\mathrm{d}x&\int_{0}^{1}x^5\,\mathrm{d}x &\int_{0}^{1}x^6\,\mathrm{d}x&\int_{0}^{1}x^3e^x\,\mathrm{d}x\\[4mm]
\int_{0}^{1}xe^x\,\mathrm{d}x&\int_{0}^{1}x^2e^x\,\mathrm{d}x&\int_{0}^{1}x^3e^x\,\mathrm{d}x&\int_{0}^{1}e^{2x}\,\mathrm{d}x
\end{vmatrix}<\frac{e^2-1}{210}\end{equation*}
\begin{equation*}\left|\begin{array}{lll}
\int_{-1}^{1}x^2\,\mathrm{d}x&\int_{-1}^{1}(x^3+2x^3\sin x)\,\mathrm{d}x &\int_{-1}^{1}(x^4+2x^4\sin^2x)\,\mathrm{d}x \\[4mm]
\int_{-1}^{1}(x^3-2x^3\sin x)\,\mathrm{d}x&\int_{-1}^{1}x^4\,\mathrm{d}x &\int_{-1}^{1}(x^5+2x^5\sin^3x)\,\mathrm{d}x\\[4mm]
\int_{-1}^{1}(x^4-2x^4\sin^2x)\,\mathrm{d}x&\int_{-1}^{1}(x^5-2x^5\sin^3x)\,\mathrm{d}x&\int_{-1}^{1}x^6\,\mathrm{d}x
\end{array}\right|>\frac{32}{2625}\end{equation*}