Question

Интеграл от -1 до 0 xe^3x dx

Answer 1

$\displaystyle\\\int\limits^0_{-1} {xe^{3x}} \, dx=\{u=x ,\ dv=e^{3x};\ \ du=dx,\ v=\frac{e^{3x}}{3}\}=x*\frac{e^{3x}}{3}-\int \frac{e^{3x}}{3}dx=\\\\\\= x*\frac{e^{3x}}{3}-\frac{1}{3}\int e^{3x}dx=x*\frac{e^{3x}}{3}-\frac{1}{3}*\frac{1}{3}e^{3x}=\bigg(\frac{xe^{3x}}{3}-\frac{e^{3x}}{9} \bigg)\mid ^0_{-1}=\\\\\\=\frac{0e^{3*0}}{3}-\frac{e^{3*0}}{9}-\bigg(\frac{-1e^{3*(-1)}}{3}-\frac{e^{3*(-1)}}{9} \bigg)=-\frac{1}{9}+\frac{4}{9e^3}$