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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


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

Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths and Science at Teachoo.