Question Number 9821 by ridwan balatif last updated on 06/Jan/17
Commented by prakash jain last updated on 08/Jan/17
$$\mathrm{Is}\:\mathrm{the}\:\mathrm{question} \\ $$$$\mathrm{Given}\:{f}\left({x}\right)={f}\left({x}+\mathrm{2}\right) \\ $$$$\mathrm{and}\:\int_{\mathrm{0}} ^{\mathrm{2}} {f}\left({x}\right){dx}={B} \\ $$$$\mathrm{then}\:\mathrm{find}\:\int_{\mathrm{3}} ^{\mathrm{7}} {f}\left({x}\right){dx} \\ $$$$\int_{\mathrm{3}} ^{\mathrm{7}} {f}\left({x}\right){dx}=\int_{\mathrm{3}} ^{\mathrm{5}} {f}\left({x}\right){dx}+\int_{\mathrm{5}} ^{\mathrm{7}} {f}\left({x}\right){dx} \\ $$$$=\mathrm{2}\int_{\mathrm{3}} ^{\mathrm{5}} {f}\left({x}\right){dx} \\ $$$$\mathrm{2}\int_{\mathrm{3}} ^{\mathrm{5}} {f}\left({x}\right){dx}=\mathrm{2}\left(\int_{\mathrm{3}} ^{\mathrm{4}} {f}\left({x}\right){dx}+\int_{\mathrm{4}} ^{\mathrm{5}} {f}\left({x}\right){dx}\right) \\ $$$$=\mathrm{2}\left(\int_{\mathrm{3}} ^{\mathrm{4}} {f}\left({x}\right){dx}+\int_{\mathrm{2}} ^{\mathrm{3}} {f}\left({x}\right){dx}\right)=\mathrm{2}\int_{\mathrm{2}} ^{\mathrm{4}} {f}\left({x}\right){dx}= \\ $$$$=\mathrm{2}{B} \\ $$