# Compare values of the three definite integrals given $f(x)=(\tan x)^\frac32-3\tan x+\sqrt{\tan x}$

Let $$f(x)=(\tan x)^\frac32-3\tan x+\sqrt{\tan x}$$. Consider the 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, prove that $$I_1>I_3>I_2$$

I found $$f'(x)$$ but it is not very straight-forward on how to determine the exact nature of the function. My idea to compare the area under curve rather than explicitly evaluate the integral.

A similar question is here but there are no solutions.

You remember the graph of $$\tan x$$? The one that is flat near zero but rapidly zooms off to infinity around $$\frac\pi2 \approx 1.57$$? Maybe the area under the curve around $$x=0$$ will be less than the area under curve around the region $$x=1$$!

Hint upon hint.

Notice also that $$f(x)$$ is in terms of positive $$\tan x$$ only, so we don't have to fanangle too much.

Lastly, notice that the bounds of the integrals have a difference of 1.

This should allow you to reason out the answer with out any messy number. Only visualization.

Comment any questions, and have fun mathing!

• I dont quite understand this statement of yours :"Maybe the area under the curve around $x=0$ will be less than the area under curve around the region $x=1$!" . Specifically, what do you mean when using the word "around" ? Do you mean some neighborhood about the point ? If so, this becomes much unreasonable, as there seems to be no reason to observe neighborhoods. I think this needs a circumstantial clarification! Commented Apr 28, 2023 at 14:32
• @FdstZfsy Looking back at my answer, I'd say I am very wrong in my train of thought, and could've phrased myself better. My point with neighborhoods was the area of a region around x=0 would be smaller than around x=1, but it is irrelevant for the question because I didn't account for the -3tan(x) term. I wonder if a better approach might be to define a function $I(n) = \int_n^{n+1}$, take the derivative, see if it is increasing or decreasing, and apply it logically to the given function. Thank you for pointing out my error though, I appreciate it! Commented Apr 29, 2023 at 7:01