How to prove that
∫∞0sin4(x)ln(x)⋅dxx2=π4⋅(1−γ).
Here is my attempt:
I(a)=∫∞0ln(x)sin4(x)xadx
I′(a)=∫∞0sin4(x)xadx
I′(2)=∫∞0sin4(x)x2dx
I′(2)=∫∞0sin2(x)x2dx−14∫∞0sin2(2x)x2dx=−π2
Why is this way wrong?
How to prove (1)?
Answer
This is solved essentially in the same way explained in answers to your previous question. As a convenient starting point, I will refer to @Jack D'Aurizio's answer:
∫∞01−cos(kx)x2log(x)dx=kπ2(1−γ−logk).
Now all you have to do is to write
sin4x=12(1−cos(2x))+18(1−cos(4x)).
I hope that the remaining computation is clear to you.
For your attempt, a correct computation would begin with
dda∫∞0sin4xxadx=−∫∞0sin4xxalogxdx.
Notice that you misidentified the derivative of your parametrized integral.
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