Question

Resolucion de integral indefinida metodo por partes .

Answer 1

$\displaystyle I=\int e^x\cos x~dx\\ \\ \\ \texttt{Sabemos bien que }\frac{d(e^x)}{dx}=e^x\texttt{ es decir }de^x=e^xdx\\ \texttt{por ende}\\ \\ I=\int \cos x~de^x\\ \\ \\ \texttt{Luego recordemos que }\int u~dv=uv-\int v~du\texttt{ as\'i tenemos}\\ \\ I=e^x\cos x-\int e^x~d(\cos x)\\ \\ I=e^x\cos x-\int e^x(-\sin x)~dx\\ \\ I=e^x\cos x+\int e^x\sin x~dx\\ \\ I=e^x\cos x+\underbrace{\boxed{\int \sin x~de^x}}_{\texttt{el mismo caso aqu\'i}}$


$\displaystyle I=e^x\cos x+\left[e^x\sin{x}-\int e^x~d(\sin x)\right]\\ \\ \\ I=e^x\cos x+\left[e^x\sin{x}-\underbrace{\int e^x\cos x~dx}_{=I}\right]\\ \\ \\ I=e^x\cos x+\left[e^x\sin{x}-I\right]\\ \\ \\ 2I=e^x\cos x+e^x\sin{x}\\ \\ \\ I=\dfrac{1}{2}e^x\cos x+\dfrac{1}{2}e^x\sin{x}+C\\ \\ \\ \boxed{\boxed{I=\dfrac{1}{2}e^x(\sin x+\cos x)+C}}$