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Ex 7.4, 20 - Integrate x + 2 / root 4x - x2 - Class 12 - Integration by specific formulaes - Method 10

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Ex 7.4, 20 (π‘₯ + 2)/√(4π‘₯ βˆ’ π‘₯^2 ) ∫1β–’(π‘₯ + 2)/√(4π‘₯ βˆ’ π‘₯^2 )=βˆ’ 1/2 ∫1β–’(βˆ’2π‘₯ βˆ’ 4)/√(4π‘₯ βˆ’ π‘₯^2 ) =βˆ’ 1/2 ∫1β–’(βˆ’2π‘₯ + 4 βˆ’ 4 βˆ’ 4)/√(4π‘₯ βˆ’ π‘₯^2 ) 𝑑π‘₯ =βˆ’ 1/2 ∫1β–’(βˆ’2π‘₯ + 4 βˆ’ 8)/√(4π‘₯ βˆ’ π‘₯^2 ) 𝑑π‘₯ =βˆ’ 1/2 ∫1β–’(βˆ’2π‘₯ + 4)/√(4π‘₯ βˆ’ π‘₯^2 ) 𝑑π‘₯+8/2 ∫1▒𝑑π‘₯/√(4π‘₯ βˆ’ π‘₯^2 ) =βˆ’ 1/2 ∫1β–’(βˆ’2π‘₯ + 4)/√(4π‘₯ βˆ’ π‘₯^2 ) 𝑑π‘₯+4∫1▒𝑑π‘₯/√(4π‘₯ βˆ’ π‘₯^2 ) Solving π‘°πŸ I1=βˆ’ 1/2 ∫1β–’(4 βˆ’ 2π‘₯)/√(4π‘₯ βˆ’ π‘₯^2 ) . 𝑑π‘₯ Let 4π‘₯ βˆ’ π‘₯^2=𝑑 Diff both sides w.r.t. x 4βˆ’2π‘₯=𝑑𝑑/𝑑π‘₯ 𝑑π‘₯=𝑑𝑑/(4 βˆ’ 2π‘₯) Thus, our equation becomes I1=βˆ’ 1/2 ∫1β–’(4 βˆ’ 2π‘₯)/√(4π‘₯ βˆ’ π‘₯^2 ) . 𝑑π‘₯ Putting the values of (4π‘₯βˆ’π‘₯^2 ) and 𝑑π‘₯ ∴ I1=βˆ’ 1/2 ∫1β–’(4 βˆ’ 2π‘₯)/βˆšπ‘‘ . 𝑑π‘₯ I1=βˆ’ 1/2 ∫1β–’(4 βˆ’ 2π‘₯)/βˆšπ‘‘ . 𝑑𝑑/(4 βˆ’ 2π‘₯) I1=βˆ’ 1/2 ∫1β–’1/βˆšπ‘‘ 𝑑𝑑 I1=βˆ’ 1/2 ∫1β–’1/(𝑑)^(1/2) 𝑑𝑑 I1=βˆ’ 1/2 〖𝑑 γ€—^((βˆ’1)/2 + 1)/((βˆ’1)/2 + 1) +𝐢1 I1=βˆ’ 1/2 (𝑑 ^(1/2 ))/(1/2) +𝐢1 I1=βˆ’γ€–π‘‘ γ€—^(1/2 )+𝐢1 I1=βˆ’βˆšπ‘‘+𝐢1 I1=βˆ’βˆš(4π‘₯βˆ’π‘₯^2 )+𝐢1 Solving π‘°πŸ I2=4∫1β–’1/√(4π‘₯ βˆ’ π‘₯^2 ) . 𝑑π‘₯ I2=4∫1β–’1/√(βˆ’(π‘₯^2 βˆ’ 4π‘₯) ) . 𝑑π‘₯ I2=4∫1β–’1/√(βˆ’(π‘₯^2 βˆ’ 2(2)(π‘₯)) ) . 𝑑π‘₯ I2=4∫1β–’1/√(βˆ’(π‘₯^2 βˆ’ 2(2)(π‘₯) + (2)^2 βˆ’ (2)^2 ) ) . 𝑑π‘₯ I2=4∫1β–’1/√(βˆ’[(π‘₯ βˆ’ 2)^2 βˆ’ (2)^2 ] ) . 𝑑π‘₯ I2=4∫1β–’1/√(βˆ’[(π‘₯ βˆ’ 2)^2 βˆ’ 4] ) . 𝑑π‘₯ I2=4∫1β–’1/√(4 βˆ’ (π‘₯ βˆ’ 2)^2 ) . 𝑑π‘₯ I2=4∫1β–’1/√((2)^2 βˆ’ (π‘₯ βˆ’ 2)^2 ) . 𝑑π‘₯ I2=4∫1β–’1/√((2)^2 βˆ’ (π‘₯ βˆ’ 2)^2 ) . 𝑑π‘₯ I2=4[γ€–sin^(βˆ’1) 〗⁑((π‘₯ βˆ’ 2)/2) ]+𝐢2 Putting values of I1 and I2 in eq. (1) ∫1β–’(π‘₯ + 2)/√(4π‘₯ βˆ’ π‘₯^2 ) . 𝑑π‘₯ =βˆ’ 1/2 ∫1β–’(4 βˆ’ 2π‘₯)/√(4π‘₯ βˆ’ π‘₯^2 ) . 𝑑π‘₯+4∫1β–’1" " /√(4π‘₯ βˆ’ π‘₯^2 ) . 𝑑π‘₯ =βˆ’βˆš(4π‘₯βˆ’π‘₯^2 )+𝐢1+4 γ€–sin^(βˆ’1) 〗⁑((π‘₯ βˆ’ 2)/2) +𝐢2 =βˆ’βˆš(πŸ’π’™βˆ’π’™^𝟐 )+πŸ’ γ€–γ€–π’”π’Šπ’γ€—^(βˆ’πŸ) 〗⁑((𝒙 βˆ’ 𝟐)/𝟐) +π‘ͺ

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Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.
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