Ex 7.2, 36 Integrate ((𝑥 + 1) (𝑥 + log⁡𝑥 )^2)/𝑥 ∫1▒〖" " ((𝑥 + 1) (𝑥 + log⁡𝑥 )^2)/𝑥〗 . 𝑑𝑥 Let 𝑥+log⁡𝑥= 𝑡 Differentiating both sides 𝑤.𝑟.𝑡.𝑥 1+1/𝑥= 𝑑𝑡/𝑑𝑥 (𝑥 + 1)/𝑥= 𝑑𝑡/𝑑𝑥 " " 𝑑𝑥 = ((𝑥 )/(𝑥 + 1))𝑑𝑡 Now, our function becomes ∫1▒〖" " ((𝑥 + 1) (𝑥 + log⁡𝑥 )^2)/𝑥〗 . 𝑑𝑥 Putting the value of 𝑥−𝑙𝑜𝑔⁡𝑥=𝑡 & 𝑑𝑥=((𝑥 )/(𝑥 + 1))𝑑𝑡 = ∫1▒〖" " ((𝑥 + 1) (𝑡)^2)/𝑥〗 . (𝑥 )/((𝑥 + 1) ) . 𝑑𝑡 = ∫1▒〖" " 𝑡^2 〗. 𝑑𝑡" " = 𝑡^(2 + 1)/(2 + 1) +𝐶 = 𝑡^3/3 +𝐶 = 𝟏/𝟑 (𝒙+𝒍𝒐𝒈⁡𝒙 )^𝟑+𝑪 (Using ∫1▒𝑥^𝑛 . 𝑑𝑥=𝑥^(𝑛+1)/(𝑛 +1)) (Using 𝑡=𝑥+𝑙𝑜𝑔⁡𝑥)

Next: Ex 7.2, 37 → Ask a doubt (live)

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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 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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