How do I solve 6 complex integrals using contours?

[nasim]

Hello ppl,


I'm trying to solve these 6 improper integrals using calculus of residues.
OK, I have actually got 7 now...


(1) $$\int_{0}^{\infty} \frac{\ln(1+x)}{1+x^{2}} dx$$
PS: I already know how to solve
$$\int_{0}^{\infty} \frac{\ln(x)}{1+x^{2}} dx$$
which equals 0, where ln(z) is a multiple-valued function
in the complex domain with branch point z=0.
But I didn't know what contour to use for (1) since
the branch point of ln(1+z) is at z=-1. If I indent it
at z=-1 and use a similar shaped contour to that of
ln(z), I get the contribution from -1 to 0 in addition to
the contribution from 0 to $$\infty$$, which
throws me off.


(2) $$\int_{0}^{\infty} \frac{\ln(1+x+x^{2})}{1+x^{2}} dx$$
PS: Again, I know how to solve
$$\int_{0}^{\infty} \frac{\ln(1+x^{2})}{1+x^{2}} dx$$
which equals $$\pi \ln 2$$, but the presence of "x"
within $$1+x+x^{2}$$ in (2) is giving me a hard time.


(3) $$\int_{0}^{\infty} \frac{\ln^{3}(1+x^{2})}{1+x^{2}} dx$$ i.e. $$\int_{0}^{\infty} \frac{(\ln(1+x^{2}))^{3}}{1+x^{2}} dx$$


(4) $$\int_{0}^{\infty} \frac{x \ln(1+x^{2})}{1+x^{2}} dx$$


(5) $$\int_{0}^{\infty} \frac{\ln(1+x^{2})}{(1+x^{2}) \sqrt{4+x^{2}}} dx$$


(6) $$\int_{0}^{\infty} \frac{\sqrt{x} \ln{(1+x)}}{1+x^{2}} dx$$
Here, the numerator consists of product of 2 multi-valued functions
with differing branch points within the complex domain
[one at z=0 for $$\sqrt{z}$$, the other at z=-1 for ln(1+z)].
How do I tackle 2 branch points and what would be the best
contour to use here?


(7) $$\int_{0}^{\infty} \frac{\sqrt{x} \sin^{-1}(1+x)}{1+x^{2}} dx$$
Again, the numerator here consists of product of 2 multi-valued functions
(sqrt and arc sin); with one branch point at z=0 for $$\sqrt{z}$$,
and then 2 branch cuts for $$\sin^{-1}(1+z)$$
(i) from $$-\infty$$ to -2, and (ii) from 0 to $$\infty$$.


Thanks,
---Nasim (nasim09021975@gmail.com)

 

[Gib Z]

Were you given these for homework or just wondered how to do them? Because the solutions I get for some of these are extremely difficult.

My solution to number 2, which agrees with Mathematica, is 8 lines long and contains non elementary functions.

 

[nasim]

Were you given these for homework or just wondered how to do them? Because the solutions I get for some of these are extremely difficult.

My solution to number 2, which agrees with Mathematica, is 8 lines long and contains non elementary functions.

wow, that is cool, Gib Z ! will you please let me know (to
the best of your ability) how you solved them, at least if I can
observe/witness/critic your way of thinking, I believe I can
learn more of the amazing and beautiful topic of residue calculus!

after all, I am here to learn, aren't I ?

 

[Gib Z]

Actually I didn't do them by hand, I probably couldn't. I am just telling you that a Computer Program that is very intelligent gets a very long answer.

 

[Eighty]

Mathematica often gives complicated answers with stuff like the LambertW function even if the answer can be simple.