Chapter 7 Class 12 Integrals
Concept wise

Ex 7.7, 7 - Integrate root 1 + 3x - x2 - Chapter 7 CBSE - Ex 7.7

Ex 7.7, 7 - Chapter 7 Class 12 Integrals - Part 2
Ex 7.7, 7 - Chapter 7 Class 12 Integrals - Part 3

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Ex 7.7, 7 √(1+3π‘₯βˆ’π‘₯2) ∫1β–’γ€–βˆš(1+3π‘₯βˆ’π‘₯^2 ) 𝑑π‘₯γ€— =∫1β–’γ€–βˆš(1βˆ’(βˆ’3π‘₯+π‘₯^2 ) ) 𝑑π‘₯γ€— =∫1β–’γ€–βˆš(1βˆ’(π‘₯^2βˆ’3π‘₯) ) 𝑑π‘₯γ€— =∫1β–’γ€–βˆš(1βˆ’[π‘₯^2βˆ’2(3/2)(π‘₯)] ) 𝑑π‘₯γ€— =∫1β–’γ€–βˆš(1βˆ’[π‘₯^2βˆ’2(3/2)(π‘₯)+(3/2)^2βˆ’(3/2)^2 ] ) 𝑑π‘₯γ€— =∫1β–’γ€–βˆš(1βˆ’[(π‘₯βˆ’3/2)^2βˆ’9/4] ) 𝑑π‘₯γ€— =∫1β–’γ€–βˆš(1βˆ’(π‘₯βˆ’3/2)^2+9/4) 𝑑π‘₯γ€— =∫1β–’γ€–βˆš(1+γ€–9/4βˆ’(π‘₯βˆ’3/2)γ€—^2 ) 𝑑π‘₯γ€— =∫1β–’γ€–βˆš(γ€–13/4βˆ’(π‘₯βˆ’3/2)γ€—^2 ) 𝑑π‘₯γ€— =∫1β–’γ€–βˆš(γ€–(√13/2)^2βˆ’(π‘₯βˆ’3/2)γ€—^2 ) 𝑑π‘₯γ€— =(π‘₯ βˆ’ 3/2)/2 √((√13/2)^2βˆ’(π‘₯βˆ’3/2)^2 )+(√13/2)^2/2 𝑠𝑖𝑛^(βˆ’1) ((π‘₯ βˆ’ 3/2))/(√13/2)+𝐢 It is of the form ∫1β–’γ€–βˆš(π‘Ž^2βˆ’π‘₯^2 ) 𝑑π‘₯=1/2 π‘₯√(π‘Ž^2βˆ’π‘₯^2 )+π‘Ž^2/2 𝑠𝑖𝑛^(βˆ’1) π‘₯/π‘Ž+𝐢〗 ∴ Replacing π‘₯ by π‘₯βˆ’3/2 and a by √13/2 , we get =((2π‘₯ βˆ’ 3)/2)/2 √(13/4βˆ’(π‘₯βˆ’3/2)^2 )+(13/4)/2 𝑠𝑖𝑛^(βˆ’1) (((2π‘₯ βˆ’ 3)/2))/(√13/2)+𝐢 =(2π‘₯ βˆ’ 3)/4 √(13/4βˆ’(π‘₯βˆ’3/2)^2 )+13/8 𝑠𝑖𝑛^(βˆ’1) (2π‘₯ βˆ’ 3)/√13+𝐢 =(2π‘₯ βˆ’ 3)/4 √(13/4βˆ’[π‘₯^2+9/4βˆ’2(π‘₯)(3/2)] )+13/8 𝑠𝑖𝑛^(βˆ’1) (2π‘₯ βˆ’ 3)/√13+𝐢 =(2π‘₯ βˆ’ 3)/4 √(13/4βˆ’π‘₯^2βˆ’9/4+3π‘₯)+13/8 𝑠𝑖𝑛^(βˆ’1) (2π‘₯ βˆ’ 3)/√13+𝐢 =(2π‘₯ βˆ’ 3)/4 √(13/4βˆ’9/4+3π‘₯βˆ’π‘₯^2 )+13/8 𝑠𝑖𝑛^(βˆ’1) (2π‘₯ βˆ’ 3)/√13+𝐢 =(πŸπ’™ βˆ’ πŸ‘)/πŸ’ √(𝟏+πŸ‘π’™βˆ’π’™^𝟐 )+πŸπŸ‘/πŸ– π’”π’Šπ’^(βˆ’πŸ) ((πŸπ’™ βˆ’ πŸ‘)/βˆšπŸπŸ‘)+π‘ͺ

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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 14 years. He provides courses for Maths, Science and Computer Science at Teachoo