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Integrate (sin^-1 x)^2 (Sine Inverse x)^2 - Chapter 7 Class 12

Ex 7.6, 10 - Chapter 7 Class 12 Integrals - Part 2
Ex 7.6, 10 - Chapter 7 Class 12 Integrals - Part 3 Ex 7.6, 10 - Chapter 7 Class 12 Integrals - Part 4


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Ex 7.6, 10 (sin^(βˆ’1)⁑π‘₯ )^2 ∫1β–’(sin^(βˆ’1)⁑π‘₯ )^2 𝑑π‘₯ Let sin^(βˆ’1)⁑π‘₯=πœƒ ∴ π‘₯=sinβ‘πœƒ Differentiating both sides 𝑀.π‘Ÿ.𝑑.π‘₯ 𝑑π‘₯/π‘‘πœƒ=cosβ‘πœƒ 𝑑π‘₯=cosβ‘πœƒ.π‘‘πœƒ Thus, our equation becomes ∫1β–’(sin^(βˆ’1)⁑π‘₯ )^2 𝑑π‘₯ = ∫1β–’πœƒ^2 . cosβ‘πœƒ.π‘‘πœƒ =πœƒ^2 ∫1β–’γ€–cosβ‘πœƒ.π‘‘πœƒγ€—βˆ’βˆ«1β–’(𝑑(πœƒ^2 )/π‘‘πœƒ ∫1β–’γ€–cosβ‘πœƒ.π‘‘πœƒγ€—)π‘‘πœƒ =πœƒ^2 sinβ‘πœƒβˆ’βˆ«1β–’γ€–2πœƒ.sinβ‘πœƒ. π‘‘πœƒγ€— =πœƒ^2 sinβ‘πœƒβˆ’2∫1β–’γ€–πœ½ π’”π’Šπ’β‘πœ½. π’…πœ½γ€— We know that ∫1▒〖𝑓(π‘₯) 𝑔⁑(π‘₯) γ€— 𝑑π‘₯=𝑓(π‘₯) ∫1▒𝑔(π‘₯) 𝑑π‘₯βˆ’βˆ«1β–’(𝑓^β€² (π‘₯) ∫1▒𝑔(π‘₯) 𝑑π‘₯) 𝑑π‘₯ Putting f(x) = ΞΈ2 and g(x) = cos ΞΈ Solving I1 ∫1β–’γ€–πœ½ π’”π’Šπ’β‘πœ½. π’…πœ½γ€— = πœƒβˆ«1β–’γ€–sinβ‘πœƒ π‘‘πœƒγ€—βˆ’βˆ«1β–’(π‘‘πœƒ/π‘‘πœƒ ∫1β–’γ€–sinβ‘πœƒ π‘‘πœƒγ€—) π‘‘πœƒ = πœƒ(βˆ’cosβ‘πœƒ )βˆ’βˆ«1β–’γ€–1.(βˆ’cosβ‘πœƒ ) γ€— π‘‘πœƒ = βˆ’πœƒ cosβ‘πœƒ+∫1β–’cosβ‘πœƒ π‘‘πœƒ = (βˆ’πœƒ cosβ‘πœƒ+sinβ‘πœƒ )+𝐢1 Now we know that ∫1▒〖𝑓(π‘₯) 𝑔⁑(π‘₯) γ€— 𝑑π‘₯=𝑓(π‘₯) ∫1▒𝑔(π‘₯) 𝑑π‘₯βˆ’βˆ«1β–’(𝑓′(π‘₯)∫1▒𝑔(π‘₯) 𝑑π‘₯) 𝑑π‘₯ Putting f(x) = ΞΈ and g(x) = sin ΞΈ Putting value of I1 in our equation ∫1β–’(sin^(βˆ’1)⁑π‘₯ )^2 𝑑π‘₯ = πœƒ^2 sinβ‘πœƒβˆ’2∫1β–’γ€–πœ½ π’”π’Šπ’β‘πœ½. π’…πœ½γ€— =πœƒ^2 sinβ‘πœƒβˆ’2(βˆ’πœƒ cosβ‘πœƒ+sinβ‘πœƒ+𝐢1) =πœƒ^2 sinβ‘πœƒ+2πœƒ cosβ‘πœƒβˆ’2 sinβ‘πœƒβˆ’2𝐢1 =πœƒ^2 sinβ‘πœƒ+2πœƒβˆš(1βˆ’sin^2β‘γ€–πœƒ γ€— )βˆ’2 sinβ‘πœƒβˆ’π‘ͺ𝟏 =πœƒ^2 sinβ‘πœƒ+2πœƒβˆš(1βˆ’sin^2β‘γ€–πœƒ γ€— )βˆ’2 sinβ‘πœƒ+π‘ͺ =(〖𝐬𝐒𝐧〗^(βˆ’πŸ)⁑𝒙 )^𝟐 𝒙+𝟐(〖𝐬𝐒𝐧〗^(βˆ’πŸ)⁑𝒙 ) √(γ€–πŸβˆ’γ€—β‘γ€–π’™^𝟐 γ€— )βˆ’πŸπ’™+π‘ͺ π‘ˆπ‘ π‘–π‘›π‘” πœƒ=sin^(βˆ’1)⁑π‘₯ & sinβ‘πœƒ=π‘₯

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.