[tex]\displaystyle \int_{0}^1\Big(e^{f(x)}+xf'(x)e^{f(x)}\Big)\, dx = \\ \\ = \int_{0}^1e^{f(x)}\, dx + \int_{0}^1xf'(x)e^{f(x)}\, dx = \\ \\ = \int_{0}^1e^{f(x)}\, dx+\int_{0}^1 x\Big(e^{f(x)}\Big)'\, dx = \\ \\ = \int_{0}^1e^{f(x)}\, dx+\Big[xe^{f(x)}\Big]\Big|_{0}^1 - \int_{0}^1x'e^{f(x)}\, dx = \\ \\ = \int_{0}^1e^{f(x)}\, dx +e^{f(1)} - \int_{0}^1e^{f(x)}\, dx= \\ \\\\ = \boxed{e^{f(1)}}[/tex]
