Check sibling questions

Example 20 - Find integral x sin-1 x / root 1-x2 dx - Examples

Example 20 - Chapter 7 Class 12 Integrals - Part 2
Example 20 - Chapter 7 Class 12 Integrals - Part 3
Example 20 - Chapter 7 Class 12 Integrals - Part 4

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Example 20 Find ∫1β–’(π‘₯ sin^(βˆ’1)⁑π‘₯)/√(1 βˆ’ π‘₯^2 ) 𝑑π‘₯ Example 20 Find ∫1β–’(π‘₯ sin^(βˆ’1)⁑π‘₯)/√(1 βˆ’ π‘₯^2 ) 𝑑π‘₯ ∫1β–’(π‘₯ sin^(βˆ’1)⁑π‘₯)/√(1 βˆ’ π‘₯^2 ) 𝑑π‘₯ Let t = 〖𝑠𝑖𝑛〗^(βˆ’1) (π‘₯) 𝑑𝑑/𝑑π‘₯=1/√(1 βˆ’ π‘₯^2 ) dt = 𝑑π‘₯/√(1 βˆ’ π‘₯^2 ) So, our equation becomes ∫1β–’(π‘₯ sin^(βˆ’1)⁑π‘₯)/√(1 βˆ’ π‘₯^2 ) 𝑑π‘₯ = ∫1β–’γ€–sin⁑〖𝑑×𝑑 〗×𝑑π‘₯/√(1 βˆ’ π‘₯^2 )γ€— = ∫1β–’γ€–sin⁑〖𝑑×𝑑 γ€— 𝑑𝑑〗 =𝑑 ∫1β–’γ€–sin⁑〖𝑑 𝑑𝑑 βˆ’ ∫1β–’(𝑑(𝑑))/𝑑𝑑〗 γ€— ∫1β–’sin⁑〖𝑑 𝑑𝑑 γ€— 𝑑𝑑 = t (βˆ’cos t) βˆ’ ∫1β–’(βˆ’cos⁑𝑑 ) 𝑑𝑑 = βˆ’ t cos t + ∫1β–’cos⁑𝑑 𝑑𝑑 = βˆ’t cos t + sin t + C 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) = t and g(x) = sin t Hence we take First function :-f(x) = t Second function :- g(x) = sin t ∫1β–’γ€–sin⁑〖𝑑×𝑑 𝑑𝑑=𝑑 ∫1β–’γ€–sin⁑〖𝑑 𝑑𝑑 βˆ’ ∫1β–’(𝑑(𝑑))/𝑑𝑑〗 γ€—γ€— γ€— ∫1β–’sin⁑〖𝑑 𝑑𝑑 γ€— 𝑑𝑑 = t (βˆ’cost) βˆ’ ∫1β–’(βˆ’cos⁑𝑑 ) 𝑑𝑑 = βˆ’ t cost + ∫1β–’cos⁑𝑑 𝑑𝑑 = βˆ’t cost + sin t + C (∫1β–’sin⁑〖π‘₯ 𝑑π‘₯=βˆ’cos⁑π‘₯ γ€— " " ) (∫1β–’cos⁑〖π‘₯ 𝑑π‘₯ = sin⁑π‘₯ γ€— " " ) t = 〖𝑠𝑖𝑛〗^(βˆ’1) (π‘₯) sin t = x sin t = x 〖𝑠𝑖𝑛〗^2 𝑑 = π‘₯^2 1 βˆ’ cos^2⁑𝑑 = π‘₯^2 γ€–π‘π‘œπ‘ γ€—^2 t = 1 βˆ’ π‘₯^2 cos t = √(1βˆ’π‘₯^2 ) Now, Hence putting the values. ∫1β–’(π‘₯ 〖𝑠𝑖𝑛〗^(βˆ’1) π‘₯)/√(1βˆ’π‘₯^2 ) 𝑑π‘₯=" βˆ’t cost + sin t + C" =π’™βˆ’βˆš(πŸβˆ’π’™^𝟐 ) γ€–π’”π’Šπ’γ€—^(βˆ’πŸ) 𝒙 +𝐂

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Davneet Singh

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.