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Question Number 93109 by Mikael_786 last updated on 10/May/20

$$\underset{0}{\stackrel{2\pi }{\int }}{cos}^{2020}\left(x\right)dx$$

Commented by prakash jain last updated on 11/May/20

$${\int }_9^{2\pi }{\cos }^{2020}xdx$$$${\int }_0^{\pi }{\cos }^{2020}xdx+{\int }_{\pi }^{2\pi }{\cos }^{2020}xdx$$$${\int }_{\pi }^{2\pi }{\cos }^{2020}xdx={\int }_0^{\pi }{\cos }^{2020}(\pi +u)du$$$$={\int }_0^{\pi }{\cos }^{2020}udu$$$${\int }_9^{2\pi }{\cos }^{2020}xdx=2{\int }_0^{\pi }{\cos }^{2020}xdx$$$$=2({\int }_0^{\pi /2}{\cos }^{2020}xdx+{\int }_{\pi /2}^{\pi }{\cos }^{2020}xdx)$$$$\mathrm{substitute}x=\frac{\pi }{2}+u\mathrm{to}\mathrm{get}$$$$=4{\int }_0^{\pi /2}{\cos }^{2020}xdx$$$$=4\times \frac{2019.2017.2015...1}{2020\cdot 2018\cdot 2016....2}\times \frac{\pi }{2}$$

Commented by Mikael_786 last updated on 10/May/20

$$thankuSir$$

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