UNKNOWN Questions

Question Number 85846 by Jakir Sarif Mondal last updated on 25/Mar/20

$$\underset{\:\mathrm{0}} {\overset{\mathrm{50}\pi} {\int}}\:\mid\:\mathrm{cos}\:{x}\:\mid{dx}\:= \\$$

Commented by jagoll last updated on 25/Mar/20

$$\mathrm{100}×\int\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\:}}\mathrm{cos}\:\mathrm{x}\:\mathrm{dx}\: \\$$$$=\:\mathrm{100}\:×\:\left[\:\mathrm{sin}\:\mathrm{x}\:\right]\underset{\:\:\mathrm{0}} {\:}^{\frac{\pi}{\mathrm{2}}} \\$$$$=\:\mathrm{100} \\$$

Commented by mathmax by abdo last updated on 25/Mar/20

$${I}\:=\sum_{{k}=\mathrm{0}} ^{\mathrm{49}} \:\:\:\int_{{k}\pi} ^{\left({k}+\mathrm{1}\right)\pi} \:\:\:\mid{cosx}\mid{dx}\:=_{{x}={k}\pi\:+{u}} \:\:\sum_{{k}=\mathrm{0}} ^{\mathrm{49}} \:\int_{\mathrm{0}} ^{\pi} \mid{cos}\left({k}\pi+{u}\right)\mid\:{du} \\$$$$=\sum_{{k}=\mathrm{0}} ^{\mathrm{49}} \:\int_{\mathrm{0}} ^{\pi} \mid{cosu}\mid\:{du}\:=\sum_{{k}=\mathrm{0}} ^{\mathrm{49}} \:\left(\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{cosu}\:{du}\:−\int_{\frac{\pi}{\mathrm{2}}} ^{\pi} \:{cosu}\:{du}\right) \\$$$$\mathrm{50}\left[{sinu}\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} −\mathrm{50}\left[{sinu}\right]_{\frac{\pi}{\mathrm{2}}} ^{\pi} \:=\mathrm{50}\:−\mathrm{50}\left(−\mathrm{1}\right)\:=\mathrm{50}\:+\mathrm{50}\:=\mathrm{100} \\$$$$\Rightarrow{I}\:=\mathrm{100} \\$$