Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class
Integration by parts
Ex 7.6, 3
Ex 7.6, 23 (MCQ)
Example 17
Ex 7.6, 1
Ex 7.6, 2 Important
Ex 7.6, 12
Example 21 Important
Ex 7.6, 21
Ex 7.6, 5 Important
Ex 7.6, 4
Ex 7.6, 6
Ex 7.6, 15
Example 18 Important
Ex 7.6, 14 Important You are here
Ex 7.6, 7 Important
Ex 7.6, 9
Ex 7.6, 8
Ex 7.6, 11
Example 20 Important
Ex 7.6, 13 Important
Ex 7.6, 22 Important
Ex 7.6, 10 Important
Example 38 Important
Integration by parts
Last updated at May 29, 2023 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 " "