CBSE Class 12 Sample Paper for 2025 Boards
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CBSE Class 12 Sample Paper for 2025 Boards
Last updated at Oct. 3, 2024 by Teachoo
Question 29 (A) Evaluate: ∫{1/(log 𝑥)−1/((log 𝑥)^2 )}𝑑𝑥; (where ├ 𝑥>1).Let I1 =∫1▒[1/(log 𝑥)−1/(log𝑥 )^2 ]𝑑𝑥 I1 = ∫1▒〖1/(log 𝑥)−∫1▒〖1/(log𝑥 )^2 .𝑑𝑥〗〗 Solving 𝐈𝟐 I2 =∫1▒𝑙𝑜𝑔(log𝑥 )𝑑𝑥 I2 =∫1▒〖𝑙𝑜𝑔(log𝑥 ).1 𝑑𝑥〗 Using by parts ∫1▒〖𝑓(𝑥) 𝑔(𝑥) 〗 𝑑𝑥=𝑓(𝑥) ∫1▒𝑔(𝑥) 𝑑𝑥−∫1▒(𝑓^′ (𝑥) ∫1▒𝑔(𝑥) 𝑑𝑥) 𝑑𝑥 Putting f(x) = log (log x) and g(x) = 1 I2=1/𝑙𝑜𝑔𝑥 ∫1▒〖1.𝑑𝑥−∫1▒[𝑑(1/𝑙𝑜𝑔𝑥)/𝑑𝑥 ∫1▒〖1.𝑑𝑥〗] 〗 𝑑𝑥 I2=1/𝑙𝑜𝑔𝑥.𝑥−∫1▒〖(−1)/(𝑙𝑜𝑔𝑥)^2 (1/𝑥).𝑥 𝑑𝑥〗 𝐈𝟐=𝒙/𝒍𝒐𝒈𝒙 +∫1▒〖𝟏/(𝒍𝒐𝒈𝒙 )^𝟐 𝒅𝒙〗 Putting the value of I2 in I1 , we get 𝐈𝟏 =𝐈𝟐−∫1▒〖𝟏/(𝒍𝒐𝒈𝒙 )^𝟐 .𝒅𝒙〗 I1 =𝑥/log𝑥 +∫1▒〖1/(log𝑥 )^2 𝑑𝑥〗−∫1▒〖1/(log𝑥 )^2 .𝑑𝑥〗 𝐈𝟏=𝒙/𝐥𝐨𝐠𝒙 +𝑪
