Let $f(x)=\int_{x}^{x+1} \sin(e^t)dt$ then which of the following statements is/are correct

Question

Let $f(x)=\int_{x}^{x+1} \sin(e^t)dt$, then which of the following statements is/are correct

(a) $\int_0^\infty f(x)dx\leq 2$

(b) $\int_0^\infty f(x)dx\gt -e$

(c) $f(x)$ has local or global maxima at $x=\ln\bigg(\dfrac{\pi}{e+1}\bigg)$

(d) $f(x)$ is continuous

I've got option (c) and (d) but I am not able to obtain (a) and (b)

This question was asked by one of my friend to me who is in 12th standard.

Please Help me to proceed further and if possible provide full solution for (a) and (b)