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Let $I=\int e^{x}(\sin x+\cos x) d x$

Let $f(x)=\sin x$

$\Rightarrow f^{\prime}(x)=\cos x$

$\therefore I=\int e^{x}\left\{f(x)+f^{\prime}(x)\right\} d x$

It is known that, $\int e^{x}\left\{f(x)+f^{\prime}(x)\right\} d x=e^{x} f(x)+\mathrm{C}$

$\therefore I=e^{x} \sin x+\mathrm{C}$