Integration Questions

Question Number 91018 by  M±th+et+s last updated on 27/Apr/20

$$\int_{\mathrm{0}} ^{\pi} \frac{{sin}\frac{\mathrm{21}{x}}{\mathrm{2}}}{{sin}\frac{{x}}{\mathrm{2}}}{dx} \\$$

Commented by mathmax by abdo last updated on 29/Apr/20

$${let}\:{take}\:{atry}\:\:{changement}\:\frac{{x}}{\mathrm{2}}\:={t}\:{give} \\$$$$\int_{\mathrm{0}} ^{\pi} \:\frac{{sin}\left(\frac{\mathrm{21}{x}}{\mathrm{2}}\right)}{{sin}\left(\frac{{x}}{\mathrm{2}}\right)}{dx}\:=\mathrm{2}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{{sin}\left(\mathrm{21}{t}\right)}{{sint}}{dt}\:\:{let}\:{w}\left({t}\right)\:=\frac{\mathrm{1}}{{sint}} \\$$$${we}\:{have}\:{w}\left({t}\right)\:=\frac{\mathrm{2}}{{e}^{{it}} −{e}^{−{it}} }\:=\frac{\mathrm{2}}{{z}−{z}^{−\mathrm{1}} }\:\:\left(\:{z}={e}^{{it}} \right) \\$$$$=\frac{\mathrm{2}{z}}{{z}^{\mathrm{2}} −\mathrm{1}}\:=−\mathrm{2}{z}\left(\frac{\mathrm{1}}{\mathrm{1}−{z}^{\mathrm{2}} }\right)\:=−\mathrm{2}{z}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:{z}^{\mathrm{2}{n}} \:=−\mathrm{2}{e}^{{it}} \:\sum_{{n}=\mathrm{0}} ^{\infty} \:{e}^{\mathrm{2}{int}} \:\Rightarrow \\$$$${I}\:=−\mathrm{4}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{sin}\left(\mathrm{21}{t}\right)\sum_{{n}=\mathrm{0}} ^{\infty} \:{e}^{\left(\mathrm{2}{n}+\mathrm{1}\right){it}} \:{dt} \\$$$$=−\mathrm{4}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{sin}\left(\mathrm{21}{t}\right){e}^{\left(\mathrm{2}{n}+\mathrm{1}\right){t}} \:{dt}\:=−\mathrm{4}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:{A}_{{n}} \\$$$${A}_{{n}} =\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{sin}\left(\mathrm{21}{t}\right)\left({cos}\left(\mathrm{2}{n}+\mathrm{1}\right){t}\:+{isin}\left(\mathrm{2}{n}+\mathrm{1}\right){t}\right){dt} \\$$$$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{sin}\left(\mathrm{21}{t}\right){cos}\left(\mathrm{2}{n}+\mathrm{1}\right){t}\:+{i}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{sin}\left(\mathrm{21}{t}\right){sin}\left(\mathrm{2}{n}+\mathrm{1}\right){t}\:{dt} \\$$$${sina}\:{cosb}\:={cos}\left(\frac{\pi}{\mathrm{2}}−{a}\right){cosb}\:=\frac{\mathrm{1}}{\mathrm{2}}\left({cos}\left(\frac{\pi}{\mathrm{2}}−{a}+{b}\right)+{cos}\left(\frac{\pi}{\mathrm{2}}−{a}−{b}\right)\right) \\$$$$=\frac{\mathrm{1}}{\mathrm{2}}\left\{{sin}\left({a}−{b}\right)+{sin}\left({a}+{b}\right)\right\}\:\Rightarrow \\$$$$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:{sin}\left(\mathrm{21}{t}\right){cos}\left(\mathrm{2}{n}+\mathrm{1}\right){t}\:{dt}\:=\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \left\{{sin}\left(\mathrm{20}−\mathrm{2}{n}\right){t}\:+{sin}\left(\mathrm{2}{n}+\mathrm{22}\right){t}\right\}{dt} \\$$$$=\frac{\mathrm{1}}{\mathrm{2}}\left[−\frac{\mathrm{1}}{\mathrm{20}−\mathrm{2}{n}}{cos}\left(\mathrm{20}−\mathrm{2}{n}\right){t}+\frac{\mathrm{1}}{\mathrm{2}{n}+\mathrm{22}}{cos}\left(\mathrm{2}{n}+\mathrm{22}\right){t}\right]_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \\$$$$=\frac{\mathrm{1}}{\mathrm{2}}\left\{−\frac{\mathrm{1}}{\mathrm{20}−\mathrm{2}{n}}{cos}\left(\mathrm{10}−{n}\right)\pi+\frac{\mathrm{1}}{\mathrm{2}{n}+\mathrm{22}}{cos}\left({n}+\mathrm{11}\right)\pi\right. \\$$$$\left.+\frac{\mathrm{1}}{\mathrm{20}−\mathrm{2}{n}}−\frac{\mathrm{1}}{\mathrm{2}{n}+\mathrm{22}}\right\}....{be}\:{continued}... \\$$