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

Awainaideannagi 2022-07-20 Answered
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.
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Answers (1)

dasse9
Answered 2022-07-21 Author has 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.

We have step-by-step solutions for your answer!

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

You might be interested in

asked 2022-08-22
Prove that ln x x 1
I need help with this proof for my real analysis class. it is on the practice sheets and we do NOT get an answer. I proved ln ( x ) < x 1 for all x > 1 by contradiction but cannot do this one.
Prove that ln ( x ) x 1 for all x > 0
i believe you need to use MVT, I cannot use the famous inequality e x > x + 1 for all x > 0
asked 2022-07-07
Growth rate of 1 / ( log ( x ) log ( x 1 ) )
Let x > 1 be a real number. Let y = 1 log ( x ) log ( x 1 )
My question: Approximately how fast does y grow (asymptotically) in terms of x? (e.g. linear, polynomial, exponential)?
asked 2022-05-18
How prove this H 2 n H n + 1 4 n > ln 2
Show that, for every positive integer n,
1 n + 1 + 1 n + 2 + + 1 2 n + 1 4 n > ln 2
I know this
lim n 1 n + 1 + 1 n + 2 + + 1 2 n = ln 2
and use this
ln ( 1 + 1 n ) < 1 n
is not useful
But for this inequality I can't. Thank you
asked 2022-01-23
Find the differential of each function.
(a)y=x2sin(4x)
dy=
(b)y=ln((1+t2))
dy=
asked 2021-02-01
log10x=2aandlog10y=b2 write 10a in terms of x
asked 2021-06-23

Examine whether the series 1=(logn)logn is convergent.

n=2

asked 2022-08-06
Why does lim N i = 1 N 1 N 1 ϵ i converge to log [ 1 ϵ ] ?
while playing around with my equations, i found that the following has to hold for my universe to be consistent:
lim N i = 1 N 1 N 1 ϵ i log [ 1 ϵ ]  for  0 < ϵ < 1
playing with numerical implementations in mathematica seem to support this by "experiment", but i just don't see why.
Anybody any ideas?