If the transformation is from R 3 → R is T { a , b...

By "standard matrix" maybe you mean the matrix with respect to the basis
$B=\{(1,0,0),(0,1,0),(0,0,1)\}$ for ${\mathbb{R}}^3$
We could rewrite $T$ this way:
$T:{\mathbb{R}}^3\to \mathbb{R}$
$T(v)=T((x,y,z))={\int }_0^{\pi }2x{e}^t+2ysin(t)+3zcos(t)dt$
$=2x{\int }_0^{\pi }{e}^{t}dt+2y{\int }_0^{\pi }sin(t)dt+3z{\int }_0^{\pi }cos(t)dt$
To write the matrix of a transformation with respect to some basis we should first apply the transformation on the basis vectors, so for example:
$T((1,0,0))=2(1){\int }_0^{\pi }{e}^{t}dt+2(0){\int }_0^{\pi }sin(t)dt+3(0){\int }_0^{\pi }cos(t)dt$
$=2{\int }_0^{\pi }{e}^{t}dt$
$=2(e^{\pi }-e^0)$
$=2(e^{\pi }-1)$
To get the matrix you should apply $T$ to the other base vectors and form the matrix with 3 rows and 1 column.
$T((0,1,0))=2{\int }_0^{\pi }sin(t)dt=4$
$T((0,0,1))=3{\int }_0^{\pi }cos(t)dt=0$
So the matrix of $T$ with respect to $B$ is: $\left(\begin{array}{c}2(e^{\pi }-1)\\ 4\\ 0\end{array}\right)$
To apply $T$ to any $(x,y,z)\in {\mathbb{R}}^3$ simply multiply:
$T((x,y,z))=\left(\begin{array}{ccc}x& y& z\end{array}\right)\left(\begin{array}{c}2(e^{\pi }-1)\\ 4\\ 0\end{array}\right)$

You find the matrix of a transformation by calculating what the transformation does to a basis of the domain of the transformation.