Playing with Integrals: Inequalities and Integrals 1

Integration, just as derivation, reveals a new approach to proving the inequalities. Let's take a detailed view on inequalities solved by or involving intgrals.

Theorem 1. If for all $x\in[a,b]$ the following inequality holds \[f(x)\geq g(x),\] then for all $x\in[a,b]$ we have \[\int^x_af(t)\,dt\geq\int^x_ag(t)\,dt.\]

I won't present a formal proof, but rather just a simple image to ilustrate the idea.

Now armed with this theorem let's solve few very basic problems.

Problem 1. Prove the following inequality $\ln(2\sin x)>\frac{1}{2}x(\pi-x)-\frac{5}{72}\pi^2,\forall x\in\left(\frac{\pi}{6},\frac{\pi}{2}\right)$

Solution. We know the very basic inequality $\tan x>x$ for $0< x <\dfrac{\pi}{2}$ . Now if we substitute $x$ with $\dfrac{\pi}{2}-x$ we'll obtain $\cot x>\dfrac{\pi}{2}-x$ . Now integrating our inequality we obtain $\int^x_{\frac{\pi}{6}}\cot t\,dt>\int^x_{\frac{\pi}{6}}\left(\dfrac{\pi}{2}-t\right)\,dt.$ Calculating we get $\ln(\sin x)-\ln\frac{1}{2}>\left(\frac{\pi x}{2}-\frac{x^2}{2}\right)-\left(\frac{\pi}{2}\cdot\frac{\pi}{6}-\frac{1}{2}\left(\frac{\pi^2}{6}\right)\right)\Leftrightarrow$ $\Leftrightarrow \ln(2\sin x)>\frac{1}{2}x(\pi-x)-\frac{5}{72}\pi^2.$

Problem 2. For $0< x <\dfrac{\pi}{2}$ prove the following inequality $\sin x \leq \frac{x(\pi-x)}{2}.$

Problem 3. For $b\geq 1$ prove the following inequality $ab\leq e^a+b(\ln b -1).$

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Comments

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This is actually a really interesting topic that I have never heard of! Thanks for sharing!

Daniel Liu - 6 years, 9 months ago

Very nice problems keep it up nicolae

abdulmuttalib lokhandwala - 6 years, 5 months ago

Sir, Can You give Solution to Problem 3,,I am not able to do it. I have done With 2nd one

Dinesh Chavan - 6 years, 7 months ago

I could only solve it by considering two cases. Case 1. If $a>x> \ln b \geq 0$ then in this interval $e^x \geq b \geq 1$ . Thus $\displaystyle \int_{\ln b}^{a} e^x \mathrm{d}x \geq \displaystyle \int_{\ln b}^{a} b \mathrm{d}x$ . On expanding we have $e^a-b \geq b(a-\ln b) \Rightarrow ab \leq e^a+ b(\ln b -1)$ . $\space$ Case 2 If $\ln b \geq x \geq 0 >a$ then in this interval $b\geq e^x$ . Thus $\displaystyle \int_{a}^{\ln b} b\mathrm{d}x \geq \displaystyle \int_{a}^{\ln b} e^x \mathrm{d}x$ . Which on expanding gives us $b(\ln b -a) \geq b- e^a \Rightarrow ab \geq e^a + b(\ln b -1)$ . PS: Let me know if you find a mistake or an easier way, all criticisms are welcomed.

Abhijeeth Babu - 6 years, 7 months ago

thanks! :)

hemang sarkar - 6 years, 3 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$