Ex 7.6, 10 - Integrate (sin-1 x)2 - Chapter 7 Class 12 - Integration by parts

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  1. Chapter 7 Class 12 Integrals
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Ex 7.6, 10 (sin^(βˆ’1)⁑π‘₯ )^2 Putting value of ΞΈ and dΞΈ ∫1β–’(πœƒ)^2 𝑑π‘₯ = ∫1β–’πœƒ^2 . cosβ‘πœƒ.π‘‘πœƒ ∫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œƒ sinβ‘πœƒ. π‘‘πœƒγ€— ∴ ∫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œƒ sinβ‘πœƒ. π‘‘πœƒγ€— Solving I1 I1 = 2∫1β–’πœƒ.sinβ‘πœƒ π‘‘πœƒ I1 = 2(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) I1 = 2(πœƒ(βˆ’cosβ‘πœƒ )βˆ’βˆ«1β–’γ€–1.(βˆ’cosβ‘πœƒ ) γ€— π‘‘πœƒ) I1 = 2(βˆ’πœƒ cosβ‘πœƒ+∫1β–’cosβ‘πœƒ π‘‘πœƒ) I1 = 2(βˆ’πœƒ cosβ‘πœƒ+sinβ‘πœƒ )+𝐢1 I1 = βˆ’2πœƒ cosβ‘πœƒ+2 sinβ‘πœƒ+𝐢1 Putting value of I1, in eq.(1) ∫1β–’(sin^(βˆ’1)⁑π‘₯ )^2 𝑑π‘₯=πœƒ^2 sinβ‘πœƒβˆ’(βˆ’2πœƒ cosβ‘πœƒ+2 sinβ‘πœƒ+𝐢1) =πœƒ^2 sinβ‘πœƒ+2πœƒ cosβ‘πœƒβˆ’2 sinβ‘πœƒβˆ’πΆ1 =πœƒ^2 sinβ‘πœƒ+2πœƒβˆš(1βˆ’sin^2β‘γ€–πœƒ γ€— )βˆ’2 sinβ‘πœƒβˆ’πΆ1 =(〖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

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