Prove

\[\displaystyle \int_0^{2\pi}{\sqrt{a^2\cos^2(t) + b^2 \sin^2(t)}\ dt} \geq \sqrt{4\pi (\pi a b + {(a-b)}^2)}\]

As Paul J. Nahin describes this inequality considering perimeter of an ellipse using isoperimetric inequality and a nice trick, he also says that he has not been able to find a proof using integration manipulations.

I also tried it myself but I could not think of any method. Try it and post your solutions. I would be glad to see them. You can also send it to the author as he mentions in his book.

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