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Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


Ex 7.9, 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 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 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.