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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Misc 30 Evaluate the definite integral ∫_0^(πœ‹/2)β–’γ€–sin⁑2π‘₯ tan^(βˆ’1)⁑(sin⁑π‘₯ ) γ€— 𝑑π‘₯ ∫_0^(πœ‹/2)β–’γ€–sin⁑2π‘₯ tan^(βˆ’1)⁑(sin⁑π‘₯ ) γ€— 𝑑π‘₯ = ∫_0^(πœ‹/2)β–’γ€–2 sin⁑π‘₯ cos⁑π‘₯ tan^(βˆ’1)⁑(sin⁑π‘₯ ) γ€— 𝑑π‘₯ Let sin⁑π‘₯=𝑑 Differentiating both sides 𝑀.π‘Ÿ.𝑑.π‘₯ cos⁑π‘₯=𝑑𝑑/𝑑π‘₯ 𝑑π‘₯=𝑑𝑑/cos⁑π‘₯ Substituting x and dx ∫1_0^(πœ‹/2)β–’γ€–2 sin⁑〖π‘₯ cos⁑〖π‘₯ γ€–π‘‘π‘Žπ‘›γ€—^(βˆ’1) (sin⁑〖π‘₯) γ€— γ€— γ€— γ€— 𝑑π‘₯ = ∫1_0^1β–’γ€–2𝑑 cos⁑〖π‘₯ γ€–π‘‘π‘Žπ‘›γ€—^(βˆ’1) (𝑑) γ€— γ€— 𝑑𝑑/π‘π‘œπ‘ π‘₯ = ∫1_0^1β–’γ€–2𝑑 γ€–π‘‘π‘Žπ‘›γ€—^(βˆ’1) 𝑑 γ€— 𝑑𝑑 = 2∫1_0^1▒〖𝑑 γ€–π‘‘π‘Žπ‘›γ€—^(βˆ’1) (𝑑) γ€— 𝑑𝑑 =2(γ€–π‘‘π‘Žπ‘›γ€—^(βˆ’1) π‘‘βˆ«1▒𝑑 𝑑𝑑 βˆ’ ∫1β–’γ€–((𝑑 (γ€–π‘‘π‘Žπ‘›γ€—^(βˆ’1) ))/𝑑𝑑 ∫1▒〖𝑑 𝑑𝑑 γ€—) γ€— 𝑑𝑑) = 2(γ€–π‘‘π‘Žπ‘›γ€—^(βˆ’1) 𝑑 (〖𝑑/2γ€—^2 )βˆ’βˆ«1β–’1/(1 + 𝑑^2 )×𝑑^2/2 𝑑𝑑) = 2(〖𝑑/2γ€—^2 γ€–π‘‘π‘Žπ‘›γ€—^(βˆ’1) π‘‘βˆ’1/2 ∫1▒𝑑^2/2 𝑑𝑑) = 𝑑^2 γ€–π‘‘π‘Žπ‘›γ€—^(βˆ’1) π‘‘βˆ’βˆ«1▒𝑑^2/(1 + 𝑑^2 ) 𝑑𝑑 Solving 𝑰_𝟏 I_1 = ∫1▒𝑑^2/(1 + 𝑑^2 ) 𝑑𝑑 Adding and Subtracting 1 in numerator. I_1 = ∫1β–’((𝑑^2 + 1 βˆ’ 1)/(𝑑^2 + 1))𝑑𝑑 I_1= ∫1β–’((𝑑^2 + 1)/(𝑑^2 + 1)βˆ’1/(𝑑^2 + 1))𝑑𝑑 I_1= ∫1β–’(1βˆ’1/(𝑑^2 + 1)) 𝑑𝑑 I_1= ∫1β–’γ€–π‘‘π‘‘βˆ’βˆ«1▒𝑑𝑑/(𝑑^2 + 1)γ€— I_1= t βˆ’ γ€–π‘‘π‘Žπ‘›γ€—^(βˆ’1) (t) Thus, our equation becomes ∴ ∫1β–’γ€–γ€–π‘‘π‘Žπ‘›γ€—^(βˆ’1) (𝑑)×𝑑 𝑑𝑑〗= 𝑑^2 γ€–π‘‘π‘Žπ‘›γ€—^(βˆ’1) 𝑑 βˆ’I_1 = 𝑑^2 γ€–π‘‘π‘Žπ‘›γ€—^(βˆ’1) π‘‘βˆ’(π‘‘βˆ’γ€–π‘‘π‘Žπ‘›γ€—^(βˆ’1) 𝑑) = 𝑑^2 γ€–π‘‘π‘Žπ‘›γ€—^(βˆ’1) π‘‘βˆ’π‘‘+γ€–π‘‘π‘Žπ‘›γ€—^(βˆ’1) 𝑑 Now, 2∫1_0^1β–’γ€–γ€–π‘‘π‘Žπ‘›γ€—^(βˆ’1) (𝑑) 𝑑 𝑑𝑑〗 =[𝑑^2 γ€–π‘‘π‘Žπ‘›γ€—^(βˆ’1) π‘‘βˆ’π‘‘+γ€–π‘‘π‘Žπ‘›γ€—^(βˆ’1) 𝑑]_0^1 =(1^2Γ—γ€–π‘‘π‘Žπ‘›γ€—^(βˆ’1) 1βˆ’1+γ€–π‘‘π‘Žπ‘›γ€—^(βˆ’1) 1)βˆ’(0βˆ’0+γ€–π‘‘π‘Žπ‘›γ€—^(βˆ’1) (0)) =(πœ‹/4βˆ’ 1+πœ‹/4)βˆ’0 = 𝝅/πŸβˆ’πŸ

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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.