Tính tích phân sau: I=∫((x^3+1)e^x+2x^3)/(e^x+2)dx

Question

Tính tích phân sau:

\(I= \int _{0}^{1}\frac{\left(x^{3} +1\right)e^{x} +2x^{3} }{e^{x} +2} {\rm d}x .\)

Answer 1

\(I=\int _{0}^{1}\frac{\left(x^{3} +1\right)e^{x} +2x^{3} }{e^{x} +2} {\rm d}x\)

\(=\int _{0}^{1}\frac{x^{3} e^{x} +e^{x} +2x^{3} }{e^{x} +2} {\rm d}x =\int _{0}^{1}\frac{x^{3} \left(e^{x} +2\right)+e^{x} }{e^{x} +2} {\rm d}x  \)
\({=\int _{0}^{1}\left(x^{3} +\frac{e^{x} }{e^{x} +2} \right){\rm d}x  =\int _{0}^{1}x^{3} {\rm d}x +\int _{0}^{1}\frac{e^{x} }{e^{x} +2} {\rm d}x  =\left. \frac{x^{4} }{4} \right|_{0}^{1} +\left. \ln \left|e^{x} +2\right|\right|_{0}^{1} \, } \\ {=\frac{1}{4} +\ln (e+2)-\ln 3.}\)