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Chapter 7 Class 12 Integrals
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Ex 7.6, 3 - Integrate x^2 e^x - Chapter 7 Class 12 NCERT

Ex 7.6, 3 - Chapter 7 Class 12 Integrals - Part 2
Ex 7.6, 3 - Chapter 7 Class 12 Integrals - Part 3

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Ex 7.6, 3 Integrate the function π‘₯^2 𝑒π‘₯ ∫1β–’γ€–π‘₯^2 𝑒^π‘₯ 𝑑π‘₯γ€— = π‘₯^2 ∫1▒〖𝑒π‘₯ 𝑑π‘₯γ€—βˆ’βˆ«1β–’(𝑑(π‘₯^2 )/𝑑π‘₯ ∫1▒〖𝑒π‘₯ 𝑑π‘₯γ€—) 𝑑π‘₯ = π‘₯^2. 𝑒π‘₯ βˆ’βˆ«1β–’γ€–2π‘₯ . 𝑒π‘₯γ€— 𝑑π‘₯ = π‘₯^2. 𝑒π‘₯ βˆ’2∫1▒〖𝒙 . 𝒆𝒙〗 𝒅𝒙 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) = x2 and g(x) = ex …(1) Solving I1 ∫1β–’γ€–π‘₯ 𝑒^π‘₯ 𝑑π‘₯γ€— = π‘₯∫1▒𝑒π‘₯ 𝑑π‘₯βˆ’βˆ«1β–’(𝑑π‘₯/𝑑π‘₯ ∫1▒𝑒^π‘₯ 𝑑π‘₯) 𝑑π‘₯ = π‘₯𝑒π‘₯ βˆ’βˆ«1▒𝑒π‘₯ 𝑑π‘₯ = π‘₯𝑒π‘₯ βˆ’π‘’π‘₯ 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) = x and g(x) = ex Putting value of I1 in our equation ∴ ∫1β–’γ€–π‘₯^2 𝑒π‘₯" " γ€— 𝑑π‘₯" = " π‘₯^2. 𝑒π‘₯ βˆ’2∫1▒〖𝒙 . 𝒆𝒙〗 𝒅𝒙 =π‘₯^2. 𝑒π‘₯ βˆ’2(π’™π’†π’™βˆ’π’†^𝒙 )+𝐢 =π‘₯^2. 𝑒π‘₯ βˆ’2π‘₯𝑒π‘₯+γ€–2𝑒〗^π‘₯+𝐢 =𝒆𝒙 (𝒙^πŸβˆ’πŸπ’™+𝟐)+π‘ͺ

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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.