Learn Intergation from Davneet Sir - Live lectures starting soon!

Ex 7.6

Ex 7.6, 1

Ex 7.6, 2 Important

Ex 7.6, 3

Ex 7.6, 4

Ex 7.6, 5 Important

Ex 7.6, 6

Ex 7.6, 7 Important

Ex 7.6, 8

Ex 7.6, 9

Ex 7.6, 10 Important

Ex 7.6, 11

Ex 7.6, 12

Ex 7.6, 13 Important

Ex 7.6, 14 Important You are here

Ex 7.6, 15

Ex 7.6, 16

Ex 7.6, 17

Ex 7.6, 18 Important

Ex 7.6, 19

Ex 7.6, 20 Important

Ex 7.6, 21

Ex 7.6, 22 Important

Ex 7.6, 23 (MCQ)

Ex 7.6, 24 (MCQ) Important

Chapter 7 Class 12 Integrals

Serial order wise

Last updated at Dec. 20, 2019 by Teachoo

Ex 7.6, 14 〖𝑥(log𝑥)〗^2 ∫1▒〖𝑥(log𝑥 )^2.𝑑𝑥 " " 〗 ∴ ∫1▒〖𝑥(log𝑥 )^2.𝑑𝑥〗=∫1▒〖(log𝑥 )^2 𝑥 .𝑑𝑥〗 = (log𝑥 )^2 ∫1▒〖𝑥 .〗 𝑑𝑥−∫1▒((𝑑(log𝑥 )^2)/𝑑𝑥 ∫1▒〖𝑥 .𝑑𝑥〗) 𝑑𝑥 = (log𝑥 )^2 . 𝑥^2/2−∫1▒(2(log𝑥 ) 1/𝑥 ∫1▒〖𝑥 .𝑑𝑥〗) 𝑑𝑥 Now we know that ∫1▒〖𝑓(𝑥) 𝑔(𝑥) 〗 𝑑𝑥=𝑓(𝑥) ∫1▒𝑔(𝑥) 𝑑𝑥−∫1▒(𝑓′(𝑥)∫1▒𝑔(𝑥) 𝑑𝑥) 𝑑𝑥 Putting f(x) = x and g(x) = (log x)2 = 𝑥^2/2 (log𝑥 )^2−2∫1▒〖log𝑥/𝑥 . 𝑥^2/2〗 𝑑𝑥 = 𝑥^2/2 (log𝑥 )^2−∫1▒〖𝑥 log𝑥 〗 𝑑𝑥 Solving I1 I1 = ∫1▒〖𝑥 log𝑥 〗 𝑑𝑥 ∫1▒〖𝑥 log𝑥 〗 𝑑𝑥=∫1▒(log𝑥 )𝑥 𝑑𝑥 =log𝑥 ∫1▒𝑥 𝑑𝑥−∫1▒(𝑑(log𝑥 )/𝑑𝑥 ∫1▒〖𝑥.𝑑𝑥〗)𝑑𝑥 Now we know that ∫1▒〖𝑓(𝑥) 𝑔(𝑥) 〗 𝑑𝑥=𝑓(𝑥) ∫1▒𝑔(𝑥) 𝑑𝑥−∫1▒(𝑓′(𝑥)∫1▒𝑔(𝑥) 𝑑𝑥) 𝑑𝑥 Putting f(x) = x and g(x) = log x =log𝑥 (𝑥^2/2)−∫1▒〖1/𝑥 . 𝑥^2/2. 𝑑𝑥〗 =〖𝑥^2/2 log〗〖 𝑥〗−1/2 ∫1▒〖𝑥. 𝑑𝑥〗 =〖𝑥^2/2 log〗𝑥−1/2 . 𝑥^2/2 +𝐶 =〖𝑥^2/2 𝑙𝑜𝑔〗〖 𝑥〗− 𝑥^2/4 +𝐶 Putting value of I1 in (1), ∫1▒〖𝑥(log𝑥 )^2.𝑑𝑥〗=𝑥^2/2 (log𝑥 )^2−∫1▒〖 𝒙 .𝒍𝒐𝒈𝒙 𝒅𝒙〗 =𝑥^2/2 (log𝑥 )^2−((𝑥^2 (log𝑥 ))/2 − 𝑥^2/4 +𝐶1) =𝑥^2/2 (log𝑥 )^2− (𝑥^2 (log𝑥 ))/2 + 𝑥^2/4 −𝐶1 =𝒙^𝟐/𝟐 (𝒍𝒐𝒈𝒙 )^𝟐− (𝒙^𝟐 (𝒍𝒐𝒈𝒙 ))/𝟐 + 𝒙^𝟐/𝟒+𝑪 " "