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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise

Transcript

Ex 7.8, 1 ∫1_π‘Ž^𝑏▒〖π‘₯ 𝑑π‘₯γ€— ∫1_π‘Ž^𝑏▒〖π‘₯ 𝑑π‘₯γ€— Putting π‘Ž =π‘Ž 𝑏 =𝑏 β„Ž=(𝑏 βˆ’ π‘Ž)/𝑛 𝑓(π‘₯)=π‘₯ Ex 7.8, 1 ∫1_π‘Ž^𝑏▒〖π‘₯ 𝑑π‘₯γ€— ∫1_π‘Ž^𝑏▒〖π‘₯ 𝑑π‘₯γ€— Putting π‘Ž =π‘Ž 𝑏 =𝑏 β„Ž=(𝑏 βˆ’ π‘Ž)/𝑛 𝑓(π‘₯)=π‘₯ We know that ∫1_π‘Ž^𝑏▒〖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(π‘₯)=π‘₯ 𝑓(π‘Ž)=π‘Ž 𝑓(π‘Ž+β„Ž)=π‘Ž+β„Ž 𝑓 (π‘Ž+2β„Ž)=π‘Ž+2β„Ž … 𝑓(π‘Ž+(π‘›βˆ’1)β„Ž)=π‘Ž+(π‘›βˆ’1)β„Ž Hence, our equation becomes ∴ ∫_0^π‘Žβ–’π‘₯ 𝑑π‘₯ = (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 ( π‘Ž+π‘Ž+ …+π‘Ž +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’1))) 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+(β„Ž . 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)β„Ž/2) = (π‘βˆ’π‘Ž) (π‘™π‘–π‘š)┬(π‘›β†’βˆž) ( π‘Ž+(𝑛 βˆ’ 1)(𝑏 βˆ’π‘Ž)/(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+(1βˆ’ 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’0) ((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 9 years. He provides courses for Maths and Science at Teachoo.