Question Number 85097 by jagoll last updated on 19/Mar/20
$$\underset{−\pi} {\overset{\pi} {\int}}\:\mathrm{x}^{\mathrm{2020}} \:\left(\mathrm{sin}\:\mathrm{x}+\mathrm{cos}\:\mathrm{x}\right)\:\mathrm{dx}\:=\:\mathrm{8} \\ $$$$\mathrm{find}\:\underset{−\pi} {\overset{\pi} {\int}}\:\mathrm{x}^{\mathrm{2020}} \:\mathrm{cos}\:\mathrm{x}\:\mathrm{dx}\:=\:? \\ $$
Answered by john santu last updated on 19/Mar/20
$$\Rightarrow\mathrm{8}\:=\:\underset{−\pi} {\overset{\pi} {\int}}\mathrm{x}^{\mathrm{2020}} \:\mathrm{sin}\:\mathrm{x}\:\mathrm{dx}\:+\:\underset{−\pi} {\overset{\pi} {\int}}\mathrm{x}^{\mathrm{2020}} \:\mathrm{cos}\:\mathrm{x}\:\mathrm{dx} \\ $$$$\left(\mathrm{1}\right)\:\underset{−\pi} {\overset{\pi} {\int}}\mathrm{x}^{\mathrm{2020}} \:\mathrm{sin}\:\mathrm{x}\:\mathrm{dx}\:=\:\mathrm{0}\: \\ $$$$\left(\mathrm{2}\right)\:\underset{−\pi} {\overset{\pi} {\int}}\mathrm{x}^{\mathrm{2020}} \:\mathrm{cos}\:\mathrm{x}\:\mathrm{dx}\:=\:\mathrm{2}\underset{\mathrm{0}} {\overset{\pi} {\int}}\:\mathrm{x}^{\mathrm{2020}} \:\mathrm{cos}\:\mathrm{x}\:\mathrm{dx} \\ $$$$\Rightarrow\:\mathrm{8}\:=\:\mathrm{0}\:+\mathrm{2}\:\underset{\mathrm{0}} {\overset{\pi} {\int}}\mathrm{x}^{\mathrm{2020}} \:\mathrm{cos}\:\mathrm{x}\:\mathrm{dx}\: \\ $$$$\Rightarrow\:\underset{\mathrm{0}} {\overset{\pi} {\int}}\:\mathrm{x}^{\mathrm{2020}} \:\mathrm{cos}\:\mathrm{x}\:\mathrm{dx}\:=\:\mathrm{4} \\ $$
Commented by jagoll last updated on 19/Mar/20
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{mister} \\ $$