Example 24 - Find integral root 3 - 2x - x^2 dx - Teachoo

Example 24 - Chapter 7 Class 12 Integrals - Part 2

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Example 24 Find ∫1β–’βˆš(3βˆ’2π‘₯βˆ’π‘₯^2 ) 𝑑π‘₯ ∫1β–’γ€–βˆš(3βˆ’2π‘₯βˆ’π‘₯^2 ) 𝑑π‘₯γ€— =∫1β–’βˆš(3βˆ’(2π‘₯+π‘₯^2 ) ) 𝑑π‘₯ Adding and Subtracting 1^2 =∫1β–’βˆš(3βˆ’(π‘₯^2+2π‘₯+1^2βˆ’1^2 ) ) 𝑑π‘₯ =∫1β–’βˆš(3βˆ’(π‘₯^2+2π‘₯+1^2 )+1) 𝑑π‘₯ =∫1β–’βˆš(3+1βˆ’(π‘₯^2+2π‘₯+1^2 ) ) 𝑑π‘₯ =∫1β–’βˆš(4βˆ’(π‘₯^2+2π‘₯+1^2 ) ) 𝑑π‘₯ =∫1β–’βˆš(2^2βˆ’(π‘₯+1)^2 ) 𝑑π‘₯ 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 a with 2 and π‘₯ with (π‘₯+1) =1/2 (π‘₯+1) √(2^2βˆ’(π‘₯+1)^2 )+2^2/2 𝑠𝑖𝑛^(βˆ’1) ((π‘₯ + 1))/2+𝐢 =1/2 (π‘₯+1) √(4βˆ’(π‘₯^2+1^2+2π‘₯))+2𝑠𝑖𝑛^(βˆ’1) ((π‘₯ + 1))/2+𝐢 =𝟏/𝟐 (𝒙+𝟏) √(πŸ‘βˆ’πŸπ’™βˆ’π’™^𝟐 )+𝟐 π’”π’Šπ’^(βˆ’πŸ) ((𝒙 + 𝟏))/𝟐+π‘ͺ

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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.