Answer
$$\frac{2}{3\pi }$$
Work Step by Step
Given
$$\int_{0}^{1}\sin(3\pi t)dt$$
Let $x=3\pi t\ \Rightarrow \ dx=3\pi dt $, at $t=0\to x=0$, at $t=1\to x=3\pi$, then
\begin{align*}
\int_{0}^{1}\sin(3\pi t)dt&=\frac{1}{3\pi}\int_{0}^{3\pi}\sin(x)dx\\
&=\frac{-1}{3\pi}\cos x\bigg|_{0}^{3\pi}\\
&=\frac{2}{3\pi }
\end{align*}
