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Question Number 85097 by jagoll last updated on 19/Mar/20

$$\underset{-\pi }{\stackrel{\pi }{\int }}{\mathrm{x}}^{2020}(\sin \mathrm{x}+\cos \mathrm{x})\mathrm{dx}=8$$$$\mathrm{find}\underset{-\pi }{\stackrel{\pi }{\int }}{\mathrm{x}}^{2020}\cos \mathrm{x}\mathrm{dx}=?$$

Answered by john santu last updated on 19/Mar/20

$$\Rightarrow 8=\underset{-\pi }{\stackrel{\pi }{\int }}{\mathrm{x}}^{2020}\sin \mathrm{x}\mathrm{dx}+\underset{-\pi }{\stackrel{\pi }{\int }}{\mathrm{x}}^{2020}\cos \mathrm{x}\mathrm{dx}$$$$\left(1\right)\underset{-\pi }{\stackrel{\pi }{\int }}{\mathrm{x}}^{2020}\sin \mathrm{x}\mathrm{dx}=0$$$$\left(2\right)\underset{-\pi }{\stackrel{\pi }{\int }}{\mathrm{x}}^{2020}\cos \mathrm{x}\mathrm{dx}=2\underset{0}{\stackrel{\pi }{\int }}{\mathrm{x}}^{2020}\cos \mathrm{x}\mathrm{dx}$$$$\Rightarrow 8=0+2\underset{0}{\stackrel{\pi }{\int }}{\mathrm{x}}^{2020}\cos \mathrm{x}\mathrm{dx}$$$$\Rightarrow \underset{0}{\stackrel{\pi }{\int }}{\mathrm{x}}^{2020}\cos \mathrm{x}\mathrm{dx}=4$$

Commented by jagoll last updated on 19/Mar/20

$$\mathrm{thank}\mathrm{you}\mathrm{mister}$$

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