Cho \(M=\int_{0}^{\ln 2} \frac{2 e^{3 x}+e^{2 x}-1...

C

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Ta có

\(\begin{aligned} M &=\int\limits_{0}^{\ln 2} \frac{2 e^{3 x}+e^{2 x}-1}{e^{3 x}+e^{2 x}-e^{x}+1} d x=\int\limits_{0}^{\ln 2} \frac{3 e^{3 x}+2 e^{2 x}-e^{x}-\left(e^{3 x}+e^{2 x}-e^{x}+1\right)}{e^{3 x}+e^{2 x}-e^{x}+1} d x \\ &=\int\limits_{0}^{\ln 2}\left(\frac{3 e^{3 x}+2 e^{2 x}-e^{x}}{e^{3 x}+e^{2 x}-e^{x}+1}-1\right) d x=\left.\ln \left(e^{3 x}+e^{2 x}-e^{x}+1\right)\right|_{0} ^{\ln 2}-\left.x\right|_{0} ^{\ln 2}=\ln \frac{11}{4} \Rightarrow e^{M}=\frac{11}{4} \end{aligned}\)