Chapter 7 Class 12 Integrals
Concept wise

    Slide32.JPG

Slide33.JPG
Slide34.JPG
Slide35.JPG
Slide36.JPG

Go Ad-free

Transcript

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

Davneet Singh's photo - Co-founder, Teachoo

Made by

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, Social Science, Physics, Chemistry, Computer Science at Teachoo.