Find eigenvalues for $T(f) = \int_{-\infty}^x tf(t)dt$ Let $V$ be the linear space of all functions continuous on $(-\infty, \infty)$ and such that that the integral $\int_{-\infty}^x tf(t)\,dt$ exists. If $f \in V$, let $g=T(f)$ be defined as $g(x) = \int_{-\infty}^x tf(t)\,dt$. Prove that every $\lambda < 0$ is an eigenvector and determine the eigenfunctions corresponding to $\lambda$.
We know that $T(f) = \int_{-\infty}^x tf(t)\,dt = \lambda f(x)$. (Not sure if this is right so far.) So know what do I do? 

Integral equations are pretty difficult to solve directly. On the other hand, we can easily solve differential equations. Try taking a derivative and use the fundamental theorem of calculus to get a differential equation which you can easily solve. (You should have $\lambda f(x)$, not $\lambda f(t)$ on the right hand side, by the way.)
