Let $f(x)=(\tan x)^{\frac{3}{2}}-3\tan x+\sqrt{\tan x}.$ Consider the three integrals $$ I_1=\int_0^1f(x)\ dx, I_2=\int_{0.3}^{1.3}f(x)\ dx, I_3=\int_{0.5}^{1.5}f(x)\ dx $$ Then how to show that $I_1>I_3>I_2.$
Can we solve it without solving the integral?
Hint: Observe that one of the integrals is totally negative. Also, you may gain intuition about what to actually try to prove by sketching the graph.
Below is the graph in the segment that is needed: