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

Transcript

Ex 7.9, 8 ∫_(πœ‹/6)^(πœ‹/4)β–’γ€–π‘π‘œπ‘ π‘’π‘ π‘₯〗⁑𝑑π‘₯ Let F(π‘₯)=∫1β–’γ€–π‘π‘œπ‘ π‘’π‘ π‘₯ . 𝑑π‘₯γ€— Multiplying and Dividing by π‘π‘œπ‘ π‘’π‘ π‘₯+π‘π‘œπ‘‘ π‘₯ F(π‘₯)=∫1β–’(π‘π‘œπ‘ π‘’π‘ π‘₯ (π‘π‘œπ‘ π‘’π‘ π‘₯ + π‘π‘œπ‘‘ π‘₯))/(π‘π‘œπ‘ π‘’π‘ π‘₯ + π‘π‘œπ‘‘ π‘₯) 𝑑π‘₯ Let cπ‘œπ‘ π‘’π‘ π‘₯+π‘π‘œπ‘‘ π‘₯=𝑑 Differentiating w.r.t. π‘₯ 𝑑/𝑑π‘₯ (π‘π‘œπ‘ π‘’π‘ π‘₯+π‘π‘œπ‘‘ π‘₯)=𝑑𝑑/𝑑π‘₯ βˆ’π‘π‘œπ‘ π‘’π‘^2 π‘₯βˆ’π‘π‘œπ‘ π‘’π‘ π‘₯ π‘π‘œπ‘‘ π‘₯=𝑑𝑑/𝑑π‘₯ βˆ’π‘π‘œπ‘ π‘’π‘ π‘₯(π‘π‘œπ‘ π‘’π‘ π‘₯+π‘π‘œπ‘‘ π‘₯)=𝑑𝑑/𝑑π‘₯ 𝑑π‘₯=𝑑𝑑/(βˆ’π‘π‘œπ‘ π‘’π‘ π‘₯ (π‘π‘œπ‘ π‘’π‘ π‘₯ + π‘π‘œπ‘‘ π‘₯) ) Therefore, ∫1β–’(π‘π‘œπ‘ π‘’π‘ π‘₯(π‘π‘œπ‘ π‘’π‘ π‘₯ + π‘π‘œπ‘‘ π‘₯))/(π‘π‘œπ‘ π‘’π‘ π‘₯ + π‘π‘œπ‘‘ π‘₯) 𝑑π‘₯ =∫1β–’(π‘π‘œπ‘ π‘’π‘ π‘₯(π‘π‘œπ‘ π‘’π‘ π‘₯ + π‘π‘œπ‘‘ π‘₯))/𝑑. 𝑑𝑑/(βˆ’π‘π‘œπ‘ π‘’π‘ π‘₯(π‘π‘œπ‘ π‘’π‘ π‘₯ + π‘π‘œπ‘‘ π‘₯) ) =βˆ’βˆ«1▒𝑑𝑑/𝑑 =βˆ’log⁑〖 |𝑑|γ€— =βˆ’log⁑〖 |π‘π‘œπ‘ π‘’π‘ π‘₯+π‘π‘œπ‘‘ π‘₯|γ€— Hence, F(π‘₯)=βˆ’log⁑|π‘π‘œπ‘ π‘’π‘ π‘₯+cot⁑π‘₯ | Now, ∫_(πœ‹/6)^(πœ‹/4)β–’γ€–π‘π‘œπ‘ π‘’π‘ π‘₯=𝐹(πœ‹/4)βˆ’πΉ(πœ‹/6) γ€— =βˆ’π‘™π‘œπ‘”|π‘π‘œπ‘ π‘’π‘(πœ‹/4)+π‘π‘œπ‘‘(πœ‹/4)|βˆ’ (βˆ’log⁑|π‘π‘œπ‘ π‘’π‘(πœ‹/6)+ π‘π‘œπ‘‘(πœ‹/6)| ) =βˆ’π‘™π‘œπ‘”|√2+1|+π‘™π‘œπ‘”|2+√3| =βˆ’π‘™π‘œπ‘”|2+√3|βˆ’ π‘™π‘œπ‘”|√2+1| (Put 𝑑=π‘π‘œπ‘ π‘’π‘ π‘₯+π‘π‘œπ‘‘ π‘₯) =π‘™π‘œπ‘”|(2 + √3)/(√2 + 1)| =π‘™π‘œπ‘”|(2 + √3)/(√2 + 1)Γ—(2 βˆ’ √3)/(2 βˆ’ √3)| =π‘™π‘œπ‘”[((2)^2 βˆ’ (√3)^2)/((√2 + 1) Γ— (2 βˆ’ √3) )] =π‘™π‘œπ‘”[(4 βˆ’ 3)/(√2 + 1)(2 βˆ’ √3) ] =π‘™π‘œπ‘”[(1 Γ— (√2 βˆ’ 1))/((2 βˆ’ √3)(√2 + 1) Γ—(√2 βˆ’ 1) )] =π‘™π‘œπ‘”[(√2 βˆ’ 1)/(2 βˆ’ √3)[(√2)^2βˆ’ 1^2 ] ] =π’π’π’ˆ[(√𝟐 βˆ’ 𝟏)/(𝟐 βˆ’ βˆšπŸ‘)] (log a βˆ’ log b = log π‘Ž/𝑏)

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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.