Difficult Integral Involving the ln function Please help me solve this integral! I have tried

hawwend8u

hawwend8u

Answered question

2022-05-27

Difficult Integral Involving the ln function
Please help me solve this integral! I have tried multiple different procedures for integration by parts, as well as substitution and have not come up with anything.
ln x ( ln x + 1 ) 2 d x
Thank you

Answer & Explanation

concludirgt

concludirgt

Beginner2022-05-28Added 11 answers

Setting ln x = y x = e y
ln x ( ln x + 1 ) 2 d x = y ( y + 1 ) 2 e y d y
= ( y + 1 1 ( y + 1 ) 2 ) e y d y = e y ( 1 y + 1 1 ( y + 1 ) 2 )
If f ( y ) = 1 y + 1 , f ( y ) = ?
Now,
e y [ f ( y ) + f ( y ) ] d y = f ( y ) e y d y + f ( y ) e y = d ( e y f ( y ) )
Timiavawsw9

Timiavawsw9

Beginner2022-05-29Added 1 answers

Integration by parts can be made to work
log x ( 1 + log x ) 2 = x log x 1 x ( 1 + log x ) 2 = x log x ( 1 log x + 1 ) =
x log x log x + 1 + ( x log x ) 1 log x + 1 = x log x log x + 1 + ( 1 + log x ) 1 log x + 1 =
x log x log x + 1 + x + C = x log x + 1 + C

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