Question Number 84549 by Power last updated on 14/Mar/20
Answered by TANMAY PANACEA last updated on 14/Mar/20
$$\int_{\mathrm{0}} ^{{n}\pi} \mid{sinnx}\mid{dx} \\ $$$$\int_{\mathrm{0}} ^{\pi} \mid{sinx}\mid{dx}+\int_{\pi} ^{\mathrm{2}\pi} \mid{sinx}\mid{dx}+\int_{\mathrm{2}\pi} ^{\mathrm{3}\pi} \mid{sinx}\mid{dx}+...+\int_{\left({n}−\mathrm{1}\right)\pi} ^{{n}\pi} \mid{sinx}\mid{dx} \\ $$$${now}\:{look}\:\int_{\mathrm{0}} ^{\pi} {sinxdx}=−\mid\left({cosx}\right)\mid_{\mathrm{0}} ^{\pi} =−\left(−\mathrm{1}−\mathrm{1}\right)=\mathrm{2} \\ $$$$\boldsymbol{{now}}\:\boldsymbol{{look}}\:\boldsymbol{{value}}\:\boldsymbol{{of}}\:\boldsymbol{{each}}\:\boldsymbol{{intregal}}=\mathrm{2} \\ $$$$\boldsymbol{{so}}\:\boldsymbol{{answer}}\:\boldsymbol{{is}}\:\:\boldsymbol{{n}}×\mathrm{2}=\mathrm{2}\boldsymbol{{n}} \\ $$$$\mathrm{2}×\mathrm{2018}=\mathrm{4036} \\ $$