A question about monotonicity

Question

Is

$$D(y_l)=\int_{-\infty}^{y_l}f_0(y)\mbox{d}y+\int_{y_l}^{y_u}e^{x\ln(1/L(y_l))}L(y)^{x}f_0(y)\mbox{d}y+\frac{1}{L(y_l)}\int_{y_u}^{\infty}f_0(y)\mbox{d}y$$

with

$$x=\frac{\ln(1/L(y_l))}{\ln(L(y_u)/L(y_l))}$$

monotone?

$\rightarrow L(y)$ is monotone increasing

$\rightarrow f_0(y)$ is a density function

The first and the third terms are easy to evaluate but I dont know about the second one.

Thanks in advance.