Задача №1516

\[ \int{e^{2x}}x^3dx =\left[\begin{aligned} & u=x^3;\;du=3x^2dx;\\ & dv=e^{2x}dx;\;v=\frac{e^{2x}}{2}. \end{aligned}\right] =\frac{x^3e^{2x}}{2}-\frac{3}{2}\int{x^2}e^{2x}dx =\left[\begin{aligned} & u=x^2;\;du=2xdx;\\ & dv=e^{2x}dx;\;v=\frac{e^{2x}}{2}. \end{aligned}\right]=\\ =\frac{x^3e^{2x}}{2}-\frac{3x^2e^{2x}}{4}+\frac{3}{2}\int{x}e^2xdx =\left[\begin{aligned} & u=x;\;du=dx;\\ & dv=e^{2x}dx;\;v=\frac{e^{2x}}{2}. \end{aligned}\right]=\\ =\frac{x^3e^{2x}}{2}-\frac{3x^2e^{2x}}{4}+\frac{3xe^{2x}}{4}-\frac{3}{4}\int{e^{2x}}dx =e^{2x}\cdot\left(\frac{x^3}{2}-\frac{3x^2}{4}+\frac{3x}{4}-\frac{3}{8}\right)+C \]