Upper bound in an integral with exponential

Question

I'm looking to find an upper bound on the following integral

$$\int_0^\infty K(u)S(t-u)du\,, $$

where $$ K(u) = e^{-u}(u-u^2/2), $$

and $ S(t) < C$ for some constant $C$.

Could someone help?

Answer 1

Observe the fact that since $S(t)<C$, we can use the estimate that $$\int_{0}^\infty K(u)S(t-u)du<|C|\,|\int_0^\infty e^{-u}(u-u^2/2)du|\leq|C|\int_0^\infty |K(u)|du$$

Now to finish your proof, you have to show that $\int_{0}^\infty |e^{-u}(u-u^2/2)|du$ exists and is finite. Edit: Consider the derivative of $\frac{1}{2}e^{-u}u^2$ and the limit for $u\to\infty$.