Show that \int_1^0\frac{\ln(1+x)}{x}dx=−\frac{1}{2}\int^1_0\frac{\ln x}{1−x}dx without actually evaluating both integrals

Jazmin Perry

Jazmin Perry

Answered question

2022-01-22

Show that 10ln(1+x)xdx=1201lnx1xdx without actually evaluating both integrals

Answer & Explanation

euromillionsna

euromillionsna

Beginner2022-01-23Added 16 answers

HINT:
Note that we have
1201log(x)1xdx=xx201xlog(x2)1x2dx
=01log(x)(11x11+x)dx
Can you finish now?
Kudusind

Kudusind

Beginner2022-01-24Added 11 answers

Show01ln(1+x)xdx=1201lnx1xdx
Playing around with series expansions.
I1=01ln(1+x)xdx
=01n=0(1)nxnn+1dx
=n=0(1)nxn+1(n+1)201
=n=0(1)n(n+1)2
I2=01lnx1xdx
=01ln(1x)1(1x)dx
=01ln(1x)xdx
=01n=0xnn+1dx
=n=001xnn+1dx
=n=01(n+1)2
2I1+I2=n=02(1)n1(n+1)2

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?