What is the value of integral int 10log[(1+x)1/2+(1−x)1/2]dx

hogwartsxhoe5t 2022-10-13 Answered
What is the value of integral
0 1 log [ ( 1 + x ) 1 / 2 + ( 1 x ) 1 / 2 ] d x
? I'm applying integration by parts but couldn't find any substitution after couple of steps
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Answers (2)

amilazamiyn
Answered 2022-10-14 Author has 14 answers
Let
I = 0 1 ln ( 1 + x + 1 x ) 1 d x
Integration by parts
I = log ( 1 + x + 1 x ) x | 0 1 + 1 2 0 1 1 + x 1 x 1 + x + 1 x 1 ( 1 + x 1 x ) x d x
I = ln 2 2 + 1 2 0 1 x 2 ( 1 + 1 x 2 ) 1 x 2 d x
Put x = sin θ Then d x = cos θ d θ
I = ln 2 2 + 1 2 0 π 2 sin 2 θ cos θ ( 1 + cos θ ) cos θ d θ
I = ln 2 2 + 1 2 0 π 2 ( 1 cos θ ) d θ = ln 2 2 + π 4 1 2 .
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Aarav Atkins
Answered 2022-10-15 Author has 2 answers
If x ( 0 , 1 ) we have 1 x + 1 + x = 2 + 2 1 x 2 , hence
0 1 log ( 1 + x + 1 x ) d x = 1 2 ( log 2 + 0 1 log ( 1 + 1 x 2 ) d x ) .
On the other hand,
0 1 log ( 1 + 1 x 2 ) d x = 0 π / 2 cos ( θ ) log ( 1 + cos θ ) d θ ( I B P ) = 0 π / 2 sin 2 θ 1 + cos θ d θ = 0 π / 2 ( 1 cos θ ) d θ = π 2 2
hence:
0 1 log ( 1 + x + 1 x ) d x = π 2 + 2 log 2 4 .
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