2. (4p) a) Consider $f, g: I \rightarrow R, I \subseteq R$ interval, two differentiable functions with continuous derivatives. Determine $\int \boldsymbol{\Upsilon}(x)+f(x) g^{\prime}(x) \cdot e^{g(x)} d x$.

(3p) b) Using the result from part a) if necessary, calculate $\int \frac{x^{2} \ln x-\ln x+x}{x^{2}} e^{x+\frac{1}{x}} d x, x>0$.

Petru Vlad
2. a) The function $\mathbf{f}^{\prime}+f g^{\prime} \bar{e}^{g}$ admits primitives being continuous.

$\int f^{\prime}(x)+f(x) g^{\prime}(x) \cdot e^{g(x)} d x=\int\left[f(x) e^{g(x)}\right]^{\prime} d x=f(x) e^{g(x)}+C$

b) For $f(x)=\ln x$ and $g(x)=x+\frac{1}{x}$, we apply a) We obtain $\int \frac{x^{2} \ln x-\ln x+x}{x^{2}} e^{x+\frac{1}{x}} d x=\ln x \cdot e^{x+\frac{1}{x}}+C$