Please help me solve this: (level = high school) int_(1)^(e)((1+\ln x)x)/((1+x\ln x)^2),dx

Ty Gaines

Ty Gaines

Answered question

2022-11-05

Calculate I = 1 e ( 1 + ln x ) x ( 1 + x ln x ) 2 d x
Please help me solve this: (level = high school)
Thanks:)

Answer & Explanation

x713x9o7r

x713x9o7r

Beginner2022-11-06Added 15 answers

If you book is correct, basically the only thing you can do is integrating by parts, noticing that the derivative of ( 1 + x log ( x ) ) is ( 1 + log ( x ) )
Then, setting u = x, v = 1 + ln x ( 1 + x ln x ) 2 , you have u = 1, v = 1 1 + x ln x and then
( 1 + ln x ) x ( 1 + x ln x ) 2 d x = x 1 + x ln x + d x 1 + x ln x
and the problem is, again, that there is no closed form for the last integral.
So, confirming what already said by other participants, either the book is wrong or you must go through numerical evaluations.
What I think is that the problem is instead
1 e ( 1 + ln x ) ( 1 + x ln x ) 2 d x = e 1 + e

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