Question

Integrate the function $e^x(\sin x+\cos x)$

Answer 1

Let $I=\int e^x(\sin x+\cos x)dx$
Let $f\left(x\right)=\sin x$
$\Rightarrow f^{\prime }\left(x\right)=\cos x$
$\therefore I=\int e^x\left\{f\right(x)+f^{\prime }(x\left)\right\}dx$
$=e^{x}f\left(x\right)+C=e^x\sin x+C$
$\because \int e^x\left\{f\right(x)+f^{\prime }(x\left)\right\}dx=e^{x}f\left(x\right)+C=e^{x}sinx+C$