my complex analysis is quite shaky, given in my undergrad I never have enough course slots to formally take it and I rely on my friends and lecture notes to learn most of the basis.
I also have a huge gap in topology because of that, which I planned to fill in this summer, because I need that to understand physics like GR, QFT, and also maths like real analysis, differential geometry

*meanwhile still solving that integral, and I presume your method does not involve converting sin x into the compelx exponential*

$$I(x)=\int x^2\sin(x)e^x dx$$
Let $u=\int e^{T(x)t}dx$. Then $\frac{du}{dt}=\frac{d}{dt}\int e^{T(x)t}dx=\int \frac{\partial}{\partial t}e^{T(x)t}dx=\int T'(x)e^{T(x)t}dx$

Set $t=1$ and equating integrands (forgot/don't know necessary condition that allow one to do so)

This gives (1)

$T'(x)e^{T(x)}=x^2\sin(x)e^x=I'(x)$

Integrate wrt x to get (2)

$\frac{e^{2T(x)}}{2}=I(x)+C_1$

$e^{2T(x)}=2I(x)+C_2$

$(1) \times (2)$ then integrate

$T'(x)e^{3T(x)}=2I(x)I'(x)+C_2I'(x)$

$\frac{1}{3}e^{3T(x)}=I(x)^2+C_2I(x)+C_3$