Answer:
$\int{ \sin\left({x}\right) \times {e}^{x} } \mathrm{d} x$
$\sin\left({x}\right) \times {e}^{x}-\int{ {e}^{x} \times \cos\left({x}\right) } \mathrm{d} x$
$\sin\left({x}\right) \times {e}^{x}-\int{ \cos\left({x}\right) \times {e}^{x} } \mathrm{d} x$
$\sin\left({x}\right) \times {e}^{x}-\left( \cos\left({x}\right) \times {e}^{x}-\int{ {e}^{x} \times \left( -\sin\left({x}\right) \right) } \mathrm{d} x \right)$
$\sin\left({x}\right) \times {e}^{x}-\left( \cos\left({x}\right) \times {e}^{x}+\int{ {e}^{x} \times \sin\left({x}\right) } \mathrm{d} x \right)$
$\int{ {e}^{x} \times \sin\left({x}\right) } \mathrm{d} x=\sin\left({x}\right) \times {e}^{x}-\left( \cos\left({x}\right) \times {e}^{x}+\int{ {e}^{x} \times \sin\left({x}\right) } \mathrm{d} x \right)$
$\int{ {e}^{x} \times \sin\left({x}\right) } \mathrm{d} x=\sin\left({x}\right) \times {e}^{x}-\cos\left({x}\right) \times {e}^{x}-\int{ {e}^{x} \times \sin\left({x}\right) } \mathrm{d} x$
$\int{ {e}^{x} \times \sin\left({x}\right) } \mathrm{d} x+\int{ {e}^{x} \times \sin\left({x}\right) } \mathrm{d} x=\sin\left({x}\right) \times {e}^{x}-\cos\left({x}\right) \times {e}^{x}$
$2 \times \int{ {e}^{x} \times \sin\left({x}\right) } \mathrm{d} x=\sin\left({x}\right) \times {e}^{x}-\cos\left({x}\right) \times {e}^{x}$
$\int{ {e}^{x} \times \sin\left({x}\right) } \mathrm{d} x=\frac{ \sin\left({x}\right) \times {e}^{x} }{ 2 }-\frac{ \cos\left({x}\right) \times {e}^{x} }{ 2 }$
$\int{ {e}^{x} \times \sin\left({x}\right) } \mathrm{d} x=\frac{ \sin\left({x}\right) \times {e}^{x}-\cos\left({x}\right) \times {e}^{x} }{ 2 }$
$\begin{array} { l }\frac{ \sin\left({x}\right) \times {e}^{x}-\cos\left({x}\right) \times {e}^{x} }{ 2 }+C,& C \in ℝ\end{array}$