Tích phân 4

Tính tích phân $I=\displaystyle\int_0^1\dfrac{x-e^x}{xe^x+x+1}dx$

Chú ý $(xe^x+x+1)’=xe^x+e^x+1$

Ta có 

$\begin{aligned}I=&\displaystyle\int_0^1\dfrac{x-e^x}{xe^x+x+1}dx\\ =&\int_0^1\dfrac{xe^x+x+1-(xe^x+e^x+1)}{xe^x+x+1}dx\\ =&\int_0^1dx-\int_0^1\dfrac{(xe^x+x+1)’}{xe^x+x+1}dx\\ =&x \bigg|_0^1-\ln(xe^x+x+1) \bigg|_0^1\\ =&1-\ln(e+2)\end{aligned}$