\(I=\int_{1}^{e} \frac{\ln x \sqrt[3]{2+\ln ^{2} x}}{x} d x\) là

* Hướng dẫn giải

Ta có

\(\begin{aligned} I &=\int\limits_{1}^{e} \frac{\ln x \sqrt[3]{2+\ln ^{2} x}}{x} d x=\int\limits_{1}^{e} \ln x \sqrt[3]{2+\ln ^{2} x} d(\ln x)=\frac{1}{2} \int\limits_{1}^{e}\left(2+\ln ^{2} x\right)^{\frac{1}{3}} d\left(2+\ln ^{2} x\right) \\ &=\left.\frac{3}{8} \cdot \sqrt[3]{\left(2+\ln ^{2} x\right)^{4}}\right|_{1} ^{e}=\frac{3}{8}\left[\sqrt[3]{3^{4}}-\sqrt[3]{2^{4}}\right] \end{aligned}\)