[tex]f(x)=x \cdot 3^x\\\displaystyle \int x \cdot 3^x \, dx=[/tex]

Возьмём интеграл по частям:

[tex]u= x \qquad du=dx\\dv=3^x \, dx \qquad v=\dfrac{3^x}{\ln 3}\\\displaystyle \int x \cdot 3^x \, dx=x \cdot \dfrac{3^x}{\ln 3}-\displaystyle \int \dfrac{3^x \, dx}{\ln 3}=x \cdot \dfrac{3^x}{\ln 3}-\dfrac{3^x}{\ln ^2 3}+C=\\=\dfrac{3^x}{\ln 3} \left(x-\log_3 e)+C[/tex]