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Ex 7.10, 10 - If f(x) = Integration t sin t from 0 to x, then f'(x) is

Ex 7.10, 10 - Chapter 7 Class 12 Integrals - Part 2
Ex 7.10, 10 - Chapter 7 Class 12 Integrals - Part 3

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Ex 7.10, 10 If 𝑓(π‘₯)=∫_0^π‘₯▒〖𝑑 sin⁑𝑑 𝑑𝑑〗, then 𝑓′(π‘₯) is (A) cos π‘₯+π‘₯ 𝑠𝑖𝑛 π‘₯ (B) π‘₯ 𝑠𝑖𝑛 π‘₯ (C) π‘₯ cos π‘₯ (D) sin⁑π‘₯ π‘₯ cos⁑π‘₯ If 𝑓(π‘₯)=∫_0^π‘₯▒〖𝑑 sin⁑〖𝑑 𝑑𝑑〗 γ€— Integrating by parts 𝑓(π‘₯)=∫_0^π‘₯▒〖𝑑 sin⁑〖𝑑 𝑑𝑑〗 γ€— By ILATE rule , 𝑑 is first function and sin 𝑑 is second function Using Integration by parts ∫1▒〖𝑓(𝑑) 𝑔(𝑑) 𝑑𝑑=𝑓(𝑑) ∫1▒〖𝑔(𝑑) π‘‘π‘‘γ€—γ€—βˆ’βˆ«1β–’γ€–(𝑓′(𝑑)∫1▒〖𝑔(𝑑) 𝑑𝑑) 𝑑𝑑〗〗 Take f(t) = t & g(t) = sin t Hence ∫_0^π‘₯▒〖𝑑 sin⁑〖𝑑 𝑑𝑑=[π‘‘βˆ«1β–’sin⁑〖𝑑 π‘‘π‘‘βˆ’βˆ«1β–’(𝑑/𝑑𝑑 π‘‘Γ—βˆ«1β–’sin⁑〖𝑑 𝑑𝑑〗 )𝑑𝑑〗 ]_0^π‘₯ γ€— γ€— =[𝑑×(βˆ’cos⁑𝑑 )βˆ’βˆ«1β–’(1Γ—cos⁑𝑑 )𝑑𝑑 ]_0^π‘₯ =[βˆ’π‘‘ cos⁑〖𝑑+sin⁑𝑑 γ€— ]_0^π‘₯ =βˆ’π‘₯ cos⁑〖π‘₯+sin⁑〖π‘₯βˆ’(βˆ’0Γ—cos⁑〖0+sin⁑0 γ€— )γ€— γ€— =sin⁑π‘₯βˆ’π‘₯ cos⁑π‘₯ Hence 𝑓(π‘₯)=sin⁑〖π‘₯βˆ’π‘₯ cos⁑π‘₯ γ€— Therefore 𝑓^β€² (π‘₯)=𝑑/𝑑π‘₯ (sin⁑〖π‘₯βˆ’π‘₯ cos⁑π‘₯ γ€— ) =𝑑/𝑑π‘₯ sin⁑〖π‘₯βˆ’π‘‘/𝑑π‘₯ (π‘₯ cos⁑π‘₯ )γ€— =cos⁑〖π‘₯βˆ’(π‘₯ (𝑑(cos⁑π‘₯))/𝑑π‘₯+cos⁑〖π‘₯ 𝑑π‘₯/𝑑π‘₯γ€— )γ€— =cos⁑〖π‘₯βˆ’(π‘₯(βˆ’sin⁑π‘₯ )+cos⁑π‘₯ )γ€— =cos⁑〖π‘₯+π‘₯ sin⁑〖π‘₯βˆ’cos⁑π‘₯ γ€— γ€— =π‘₯ sin⁑π‘₯ Hence, Option B is Correct

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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.