While integrating on both the sides, instead of substituting (1+logx) witht why have they splited the numerator and substituting log x as t. Even the former one does well. And the answers i get from both the method are pity different. My question is that why can`t take (1+log x) as t. Please refer the image uploaded. Hope I could hear from you at the earliest.

After you`re done integrating, you put a c which accounts for a missing constant right? Note that, even if you follow your method there`s nothing wrong with it. Assuming that you`ve already tried your method you`ll notice that the only difference between the book solution and your solution is just a 1/2. Your solution has an extra 1/2.Now, where does your 1/2 go then? (Hint: It`s included within the constant part c in the book`s solution.) Hope this helped. Please ask if you have any further queries.