Question:

Step 1: Simplify the integrand using exponent properties. We use the property $e^{a+b} = e^a \cdot e^b$ and $e^{\ln x} = x$. $$ \int e^{\ln x + x} dx = \int (e^{\ln x} \cdot e^x) dx = \int (x \cdot e^x) dx $$ Step 2: Apply integration by parts to evaluate the integral $\int x e^x dx$. We use the integration by parts formula: $\int u \, dv = uv - \int v \, du$. Let $u = x$ and $dv = e^x dx$. Then, we find $du$ and $v$: $du = dx$ $v = \int e^x dx = e^x$ Now substitute these into the integration by parts formula: $$ \int x e^x dx = x \cdot e^x - \int e^x dx $$ Step 3: Perform the remaining integration. We evaluate the integral $\int e^x dx$. $$ \int e^x dx = e^x $$ Step 4: Substitute the result back and add the constant of integration. Substitute the result from Step 3 into the expression from Step 2: $$ x e^x - \int e^x dx = x e^x - e^x + C $$ The final answer is $\boxed{x e^x - e^x + C}$.