Let $X$ be a positive random variable with CDF $F_X(x)$, and $a$ and $b$ are two positive constant (we don't know the relation between $a$ and $b$), how should I start to evaluate the following: $$ \mathbb{E}\left[\min\{X,a\} \int_X^\infty \min\{t^{-2},b\} ~dt\right]. $$ I am able to obtain $\mathbb{E}\left[\min\{X,a\}\right]$ and $\mathbb{E}\left[\int_X^\infty \min\{t^{-2},b\} ~dt\right]$.
Any hints?
$$\int_X^\infty \min(t,b)dt = \mathbb 1 [1/X^2\leq b]\int_X^\infty 1/t^2 dt + \mathbb 1 [1/X^2> b]\left[ (\frac 1 {\sqrt b} -X)\frac 1 {b^2} + \int_b^\infty 1/t^2 dt \right]\\ = \mathbb 1 [X\geq \frac 1 {\sqrt b}]\frac 1 X + \mathbb 1 [X<\frac 1 {\sqrt b}]\left[ (\frac 1 {\sqrt b} -X)\frac 1 {b^2} + \frac 1 b\right] $$ The 1/X term doesn't smell good, some relation between $a$ and $b$ would kill it.