Cho \(\int\limits^{e}_{1}(1+x\ln x){\text{d}x}=ae^2+be+c,\left(a,b,c\in\mathbb{Q}\right).\) Mệnh đề nào dưới đây đúng?
\(a+b=c\).\(a+b=-c\).\(a-b=-c\).\(a-b=c\).Hướng dẫn giải:\(\int\limits^e_1\left(1+x\ln x\right)\text{d}x=\int\limits^e_1\text{d}x+\int\limits^e_1x\ln x\text{d}x=\left(e-1\right)+\int\limits^e_1\ln x\text{d}\frac{x^2}{2}=\left(e-1\right)+\frac{x^2}{2}\ln x|^e_1-\int\limits^e_1\frac{x^2}{2}\text{d}\ln x\)\(=\left(e-1\right)+\frac{e^2}{2}-\int\limits^e_1\frac{x}{2}\text{d}x=\frac{e^2}{2}+e-1-\frac{x^2}{4}|^e_1\)
\(=\frac{e^2}{2}+e-1-\left(\frac{e^2}{4}-\frac{1}{4}\right)=\frac{e^2}{4}+e-\frac{3}{4}\)
Suy ra \(a=\frac{1}{4},b=1,c=-\frac{3}{4}\) do đó \(a-b=c.\)