Question

Integre (x+1) e^x dx

Answer 1

$\displaystyle \sf \int(x+1)e^xdx \\\\\ Usando\ a \ integra{\c c}\~ao \ por\ partes,\ temos \ que : \\\\ \int \underbrace{\sf f(x)}_{\displaystyle \sf u}\cdot \underbrace{\sf g'(x)dx}_{\displaystyle dv} = \underbrace{\sf f(x)\cdot g(x)}_{\displaystyle \sf u\cdot v} -\int \underbrace{\sf f'(x)\cdot g(x) dx }_{\displaystyle v\cdot du } \\\\\\ \int u\cdot dv = u\cdot v - \int v\cdot du$

Daí, façamos :

$\displaystyle \sf \int \sf (x+1)\cdot e^x dx \to \underbrace{\sf u = x+1}_{\displaystyle d u = 1 }\ \ ; \ dv = \underbrace{\sf e^x}_{\displaystyle \sf v = e^x }}\\\\\\ Da{\'i}}: \\\\ \int (x+1) \cdot e^x = (x+1)\cdot e^x -\int e^x dx \\\\\\ \int x \cdot e^x = (x+1)\cdot e^x -e^x +C \\\\\\ x\cdot e^x +e^x -e^x +C \\\\ x\cdot e^x +C$

Portanto

$\huge\boxed{\sf \int (x+1)\cdot e^x dx = x\cdot e^x + C }\checkmark$