Check sibling questions
Chapter 7 Class 12 Integrals (Term 2)
Concept wise

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Transcript

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 is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.