Prove the inequality $$\left|\int_0 ^{\pi/4} \frac{\tan x~dx}{3-\sin(x^2)}\right|≤ \frac{1}{4}\log_e 2.$$
I have tried many different ways to get this inequality but failed.
2026-03-25 23:38:09.1774481889
Inequality involving integrals of trigonometric functions
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For $x\in [0,\pi/4]$, we have that $\sin(x^2)\leq 1$ and $\tan(x)\geq 0$. Therefore $3-\sin(x^2)\geq 2$ and $$0\leq \int_0 ^{\pi/4} \frac{\tan(x)}{3-\sin(x^2)}\,dx\leq \int_0 ^{\pi/4} \frac{\tan(x)}{2}\,dx.$$ Can you take it from here?