Answer
$$\frac{1}{\ln 3}\left(3-\frac{2}{\ln 3}\right)$$
Work Step by Step
Given $$\int_{0}^{1} x 3^{x} d x$$
Let
\begin{align*}
u&= x\ \ \ \ \ \ \ \ \ dv=3^xdx\\
du&=dx\ \ \ \ \ \ \ \ v=\frac{1}{\ln 3} 3^x
\end{align*}
Then
\begin{align*}
\int_{0}^{1} x 3^{x} d x&=\frac{x}{\ln 3} 3^x \bigg|_{0}^{1}-\frac{1}{\ln 3}\int_{0}^{1} 3^xdx\\
&=\frac{x}{\ln 3} 3^x-\frac{1}{(\ln 3)^2} 3^x \bigg|_{0}^{1}\\
&= \frac{1}{\ln 3}\left(3-\frac{2}{\ln 3}\right)
\end{align*}
