<s>[INST] how to do this using properties of definite integrals? $$\int_1^2 ln_2(x^3+1)dx$$
which is the same as
$\int_1^2 ln_2((3)-(x^3+1))dx$
which gives $\int_1^2 ln_2(2-x^3)\ dx$
which does not help.
Or integrate this by parts, but that's really tedious.
 [/INST] Use $x^3 + 1 = (x+1)(1 - x + x^2)$.
$$\int\limits_{x=1}^2 \log_2 (x^3 + 1)\ dx = \int\limits_{x=1}^2 \log_2 (x + 1)\ dx +\int\limits_{x=1}^2 \log_2 (1 - x + x^2)\ dx$$
Then use (from integral tables): $\int \log_2 (x+1)\ dx = (x+1) \log (x+1)-x$
and
$\int \log_2 (1 - x + x^2)\ dx = \left(x-\frac{1}{2}\right) \log \left(x^2-x+1\right)-2 x+\sqrt{3} \tan ^{-1}\left(\frac{2
   x-1}{\sqrt{3}}\right)$
sum, to find the full integral is:
$$\frac{1}{6} \left(\sqrt{3} \pi +27 \log_2 (3)-6 (3+\log_2 (4))\right)$$
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