Question

U=\frac{a\pih^2a}{m} \int\limits^a_b {x} \, dx \pi d^3 x l w (x) l^4 evaluate it

Answer 1

I'm interested in integrals of the form

I(a,b)=∫

∞

0

arccot(x)⋅arccot(ax)⋅arccot(bx) dx, for a>0,b>0

It's known†Gradshteyn & Ryzhik, Table of Integrals, Series, and Products, 7th edition, page 599, (4.511) that

I(a,0)=

π2

4

[ln(1+

1

a

)+

ln(1+a)

a

].

Maple and Mathematica are also able to evaluate

I(1,1)=

3π2

4

ln2−

21

8

ζ(3).