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Ex 7.8, 1 ∫1_π‘Ž^𝑏▒〖π‘₯ 𝑑π‘₯γ€— ∫1_π‘Ž^𝑏▒〖π‘₯ 𝑑π‘₯γ€— Putting 𝒂 =π‘Ž 𝒃 =𝑏 𝒉=(𝑏 βˆ’ π‘Ž)/𝑛 𝒇(𝒙)=π‘₯ We know that ∫1_π‘Ž^𝑏▒〖𝑓(π‘₯) 𝑑π‘₯γ€— =(π‘βˆ’π‘Ž) (π‘™π‘–π‘š)┬(π‘›β†’βˆž) 1/𝑛 (𝑓(π‘Ž)+𝑓(π‘Ž+β„Ž)+𝑓(π‘Ž+2β„Ž)…+𝑓(π‘Ž+(π‘›βˆ’1)β„Ž)) Hence we can write ∫1_π‘Ž^𝑏▒〖π‘₯ 𝑑π‘₯γ€— =(π‘βˆ’π‘Ž) lim┬(nβ†’βˆž) 1/𝑛 (𝑓(π‘Ž)+𝑓(π‘Ž+β„Ž)+𝑓(π‘Ž+2β„Ž)+… +𝑓(π‘Ž+(π‘›βˆ’1)β„Ž) Here, 𝒇(𝒙)=π‘₯ 𝒇(𝒂)=π‘Ž 𝒇(𝒂+𝒉)=π‘Ž+β„Ž 𝒇 (𝒂+πŸπ’‰)=π‘Ž+2β„Ž … 𝒇(𝒂+(π’βˆ’πŸ)𝒉)=π‘Ž+(π‘›βˆ’1)β„Ž Hence, our equation becomes ∴ ∫_𝟎^𝒂▒𝒙 𝒅𝒙 = (π‘βˆ’π‘Ž) (π‘™π‘–π‘š)┬(π‘›β†’βˆž) 1/𝑛 (𝑓(π‘Ž)+𝑓(π‘Ž+β„Ž)+𝑓(π‘Ž+2β„Ž)…+𝑓(π‘Ž+(π‘›βˆ’1)β„Ž)) = (π‘βˆ’π‘Ž) (π‘™π‘–π‘š)┬(π‘›β†’βˆž) 1/𝑛 (π‘Ž+(π‘Ž+β„Ž)+(π‘Ž+2β„Ž)+ …+(π‘Ž+(π‘›βˆ’1)β„Ž)) = (π‘βˆ’π‘Ž) (π‘™π‘–π‘š)┬(π‘›β†’βˆž) 1/𝑛 ( 𝒂+𝒂+ …+𝒂 +β„Ž+2β„Ž+ ……+(π‘›βˆ’1)β„Ž) = (π‘βˆ’π‘Ž) (π‘™π‘–π‘š)┬(π‘›β†’βˆž) 1/𝑛 ( 𝒏𝒂 +β„Ž+2β„Ž+ ……+(π‘›βˆ’1)β„Ž) = (π‘βˆ’π‘Ž) (π‘™π‘–π‘š)┬(π‘›β†’βˆž) 1/𝑛 ( π‘›π‘Ž+β„Ž (𝟏+𝟐+ ………+(π’βˆ’πŸ))) 𝒏 π’•π’Šπ’Žπ’†π’” We know that 1+2+3+ ……+𝑛= (𝑛 (𝑛 + 1))/2 1+2+3+ ……+π‘›βˆ’1= ((𝑛 βˆ’ 1) (𝑛 βˆ’ 1 + 1))/2 = (𝒏 (𝒏 βˆ’ 𝟏) )/𝟐 = (π‘βˆ’π‘Ž) (π‘™π‘–π‘š)┬(π‘›β†’βˆž) 1/𝑛 ( π‘›π‘Ž+(𝒉 . 𝒏(𝒏 βˆ’ 𝟏))/𝟐) = (π‘βˆ’π‘Ž) (π‘™π‘–π‘š)┬(π‘›β†’βˆž) ( π‘›π‘Ž/𝒏+𝑛(𝑛 βˆ’ 1)β„Ž/2𝒏) = (π‘βˆ’π‘Ž) (π‘™π‘–π‘š)┬(π‘›β†’βˆž) ( π‘Ž+(𝑛 βˆ’ 1)𝒉/2) = (π‘βˆ’π‘Ž) (π‘™π‘–π‘š)┬(π‘›β†’βˆž) ( π‘Ž+(𝑛 βˆ’ 1)(𝒃 βˆ’π’‚)/(2 . 𝒏)) = (π‘βˆ’π‘Ž) (π‘™π‘–π‘š)┬(π‘›β†’βˆž) ( π‘Ž+(𝒏/𝒏 βˆ’ 𝟏/𝒏) ((𝑏 βˆ’ π‘Ž) )/2) [π‘ˆπ‘ π‘–π‘›π‘” β„Ž=(𝑏 βˆ’ π‘Ž)/𝑛] = (π‘βˆ’π‘Ž) (π‘™π‘–π‘š)┬(π‘›β†’βˆž) ( π‘Ž+(πŸβˆ’ 𝟏/𝒏) ((𝑏 βˆ’ π‘Ž) )/2) = (π‘βˆ’π‘Ž)( π‘Ž+(1βˆ’ 𝟏/∞) ((𝑏 βˆ’ π‘Ž) )/2) = (π‘βˆ’π‘Ž)( π‘Ž+(1βˆ’πŸŽ) ((𝑏 βˆ’ π‘Ž) )/2) = (π‘βˆ’π‘Ž)( π‘Ž+ (𝑏 βˆ’ π‘Ž )/2) = (π‘βˆ’π‘Ž)((2π‘Ž + 𝑏 βˆ’ π‘Ž )/2) = (𝑏 βˆ’ π‘Ž)(𝑏 + π‘Ž)/2 = (𝒃^𝟐 βˆ’ 𝒂^𝟐)/𝟐

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.