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Prove this integral inequality involving pi

Source: 2019 Jozsef Wildt International Math Competition

May 20, 2020
integrationinequalitiescalculus

Problem Statement

Prove that ∫0π2(cos⁡x)1+2n+1dx≤2n−1n!π2(2n+1)!\int \limits_0^{\frac{\pi}{2}}(\cos x)^{1+\sqrt{2n+1}}dx\leq \frac{2^{n-1}n!\sqrt{\pi}}{\sqrt{2(2n+1)!}}for all n∈N∗n\in \mathbb{N}^*
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