Question Number 52949 by gunawan last updated on 15/Jan/19
$$\mathrm{If}\:{f}\left({x}\right)\:\mathrm{is}\:\mathrm{an}\:\mathrm{odd}\:\mathrm{function},\:\mathrm{then} \\ $$$$\underset{\:\mathrm{0}} {\overset{\pi} {\int}}\:{f}\:\left(\mathrm{cos}\:{x}\right)\:{dx}\:=\:\mathrm{2}\underset{\:\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\:{f}\:\left(\mathrm{cos}\:{x}\right)\:{dx} \\ $$
Commented by tanmay.chaudhury50@gmail.com last updated on 15/Jan/19
$${re}\:{check}\:{the}\:{question}… \\ $$
Commented by maxmathsup by imad last updated on 15/Jan/19
$$\int_{\mathrm{0}} ^{\pi} {f}\left({cosx}\right){dx}\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {f}\left({cosx}\right){dx}\:+\int_{\frac{\pi}{\mathrm{2}}} ^{\pi} {f}\left({cosx}\right){dx}\:{but} \\ $$$$\int_{\frac{\pi}{\mathrm{2}}} ^{\pi} {f}\left({cosx}\right){dx}\:=_{{x}\:=\frac{\pi}{\mathrm{2}}+{t}} \:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {f}\left(−{sint}\right){dt}\:=−\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {f}\left({sint}\right){dt}\:\left({f}\:{is}\:{odd}\right) \\ $$$$=−\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {f}\left({cos}\left(\frac{\pi}{\mathrm{2}\:}−{t}\right)\right){dt}\:=_{\frac{\pi}{\mathrm{2}\:\:}−{t}\:={u}} \:\:−\int_{\frac{\pi}{\mathrm{2}}} ^{\mathrm{0}} {f}\left({cosu}\right)\left(−{du}\right) \\ $$$$=−\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {f}\left({cosu}\right){du}\:\Rightarrow\int_{\mathrm{0}} ^{\pi} {f}\left({cosx}\right){dx}\:=\mathrm{0}\: \\ $$$${if}\:{f}\:{is}\:{even}\:{we}\:{get}\:\int_{\frac{\pi}{\mathrm{2}}} ^{\pi} \:{f}\left({cost}\right){dt}\:=_{{x}=\frac{\pi}{\mathrm{2}}+{t}} \:\:\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {f}\left(−{sint}\right){dt}\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {f}\left({sint}\right){dt} \\ $$$$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {f}\left({cos}\left(\frac{\pi}{\mathrm{2}}−{t}\right)\right){dt}\:=_{\frac{\pi}{\mathrm{2}}−{t}={u}} \:\:−\:\int_{\frac{\pi}{\mathrm{2}}} ^{\mathrm{0}} \:{f}\left({cosu}\right){du}=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {f}\left({cosu}\right){du}\:\Rightarrow \\ $$$$\int_{\mathrm{0}} ^{\pi} {f}\left({cosx}\right){dx}\:=\mathrm{2}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {f}\left({cosx}\right){dx}\:{so}\:{there}\:{is}\:{a}\:{error}\:{in}\:{the}\:{question}!… \\ $$
Commented by gunawan last updated on 16/Jan/19
$$\mathrm{thank}\:\mathrm{you}\:\mathrm{Sir} \\ $$
Commented by maxmathsup by imad last updated on 16/Jan/19
$${you}\:{are}\:{welcome} \\ $$