Calculate int_0^1(log^2x log(1+x^2))/(1-x^2)dx

Awainaideannagi

Awainaideannagi

Answered question

2022-07-20

Calculate 0 1 log 2 x log ( 1 + x 2 ) 1 x 2 d x
I found π 4 32 + 2 G 2 + 7 4 ζ ( 3 ) log 2 where G is the Catalan's constant.

Answer & Explanation

dasse9

dasse9

Beginner2022-07-21Added 12 answers

J = 0 1 log 2 ( x ) log ( 1 + x 2 ) 1 x 2 d x = IBP [ ( 0 x ln 2 t 1 t 2 d t ) ln ( 1 + x 2 ) ] 0 1 0 1 2 x 1 + x 2 ( 0 x ln 2 t 1 t 2 d t ) d x = 7 4 ζ ( 3 ) ln 2 0 1 0 1 2 x 2 ln 2 ( t x ) ( 1 + x 2 ) ( 1 t 2 x 2 ) d t d x = 7 4 ζ ( 3 ) ln 2 + 2 0 1 0 1 ( ln 2 ( t x ) ( 1 + t 2 ) ( 1 + x 2 ) ln 2 ( t x ) ( 1 + t 2 ) ( 1 t 2 x 2 ) ) d t d x = 7 4 ζ ( 3 ) ln 2 + 4 0 1 0 1 0 1 ln 2 x ( 1 + t 2 ) ( 1 + x 2 ) d t d x + 4 ( 0 1 ln x 1 + x 2 d x ) 2 2 0 1 0 1 ln 2 ( t x ) ( 1 + t 2 ) ( 1 t 2 x 2 ) d t d x = 7 4 ζ ( 3 ) ln 2 + 4 × 1 16 π 3 × 1 4 π + 4 G 2 0 1 2 t ( 1 + t 2 ) ( 0 t ln 2 u 1 u 2 d u ) d t = IBP 7 4 ζ ( 3 ) ln 2 + 1 16 π 4 + 4 G 2 [ ( 2 ln t ln ( 1 + t 2 ) ) ( 0 t ln 2 u 1 u 2 d u ) ] 0 1 + 0 1 ( 2 ln t ln ( 1 + t 2 ) ) ln 2 t 1 t 2 d t = 7 4 ζ ( 3 ) ln 2 + 1 16 π 4 + 4 G 2 + 7 4 ζ ( 3 ) ln 2 + 2 0 1 ln 3 t 1 t 2 d t J J = 7 4 ζ ( 3 ) ln 2 + 2 G 2 1 32 π 4
NB: i assume that,
0 1 ln t 1 + t 2 d t = G 0 1 ln 2 t 1 + t 2 d t = π 3 16 0 1 ln 2 t 1 t 2 d t = 7 4 ζ ( 3 ) 0 1 ln 3 t 1 t 2 d t = π 4 16
Addendum: typo fixed. Thanks Sewer Keeper.

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