Integrate using matrix representations

Has it ever occurred to you that we can integrate using matrix representations? If not, prepare to have your mind blown.

It works as follows: invert the matrix representation of the differentiation operator with respect to a clever choice of a basis and then apply the inverse of the operator to the function we wish to integrate.

Here's an example. Say we want to evaluate $$\int e^{ax}\cos bx\;dx.$$ Consider the basis $B=\{e^{ax}\cos bx,e^{ax}\sin bx\}.$ Now differentiate each basis element with respect to $x$: $$\dfrac{d}{dx}e^{ax}\cos bx=ae^{ax}\cos bx-be^{ax}\sin{bx}\\ \dfrac{d}{dx}e^{ax}\sin bx=ae^{ax}\sin bx+be^{ax}\cos{bx}.$$ The matrix representation of the differential operator is thus $$T=\begin{pmatrix}a & b\\ -b & a\end{pmatrix}.$$ To evaluate the given integral, we can calculate $$T^{-1}\begin{pmatrix}1\\0\end{pmatrix}=\dfrac{1}{a^2+b^2}\begin{pmatrix}a\\b \end{pmatrix}_B$$ As a result, $$\int e^{ax}\cos bx\;dx=\dfrac{a}{a^2+b^2}e^{ax}\cos bx+\dfrac{b}{a^2+b^2}e^{ax}\sin bx.$$ Note that there are elementary ways of evaluating this integral, but isn't this method amazing?