Задача №1440

\[ \int\frac{e^{2x}dx}{\sqrt[4]{e^x+1}} =\int\frac{e^x\cdot{e^xdx}}{\sqrt[4]{e^x+1}} =\left[\begin{aligned}& u=e^x+1;\\& du=e^xdx.\end{aligned}\right] =\int\frac{(u-1)du}{\sqrt[4]{u}}=\\ =\int\left(u^{\frac{3}{4}}-u^{-\frac{1}{4}}\right)du =\frac{4u^{\frac{7}{4}}}{7}-\frac{4u^{\frac{3}{4}}}{3}+C =4u^{\frac{3}{4}}\cdot\left(\frac{u}{7}-\frac{1}{3}\right)+C=\\ =4\sqrt[4]{\left(e^x+1\right)^3}\cdot\left(\frac{e^x+1}{7}-\frac{1}{3}\right)+C =4\sqrt[4]{\left(e^x+1\right)^3}\cdot\left(\frac{e^x}{7}-\frac{4}{21}\right)+C \]