Chapter 7 Class 12 Integrals
Concept wise

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Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


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Question 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 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.