Suppose a random variable $x$ with p.d.f $f(x) = \lambda e^{-\lambda x}$ such that $\lambda$ is the parameter of $f$. Given a function $ g(x) = (a + bx )e^{- \frac{\lambda x}{a}} $ where $a,b \in \mathbb{R}$, I would like to simplify $$ \int_0^{\infty}g(x)f(x)dx = \int_0^{\infty}g(x)\lambda e^{-\lambda x}dx $$
I do not know how to integrate this expression.