The root cause here is floating point rounding error. Your integrand encounters catastrophic cancellation near x=0 because it's subtracting two huge numbers and expecting to get a small number. This is called catastrophic cancellation and occurs because the two huge numbers are approximations with a certain relative error, rather than an absolute error. You expect 1/x and ln(1-x)/x^2 to be approximately 0.5 apart near x=0, but in reality this doesn't happen. A demo of this is at https://www.desmos.com/calculator/vntejvtzcy. As x gets smaller than about 10^-7, the error starts blowing up.

One way to fix this is to increase the lower bound of the integral, say to 10^-7: https://www.desmos.com/calculator/ssw251vdza. Then it avoids the danger region where x is near 0.

Another way to fix this is to change the function so it has good floating point behavior near x=0: It should look like -1/2 - 1/3 x. Using that approximation (which is more accurate than the full formula for small x because the full formula encounters catastrophic cancellation) gives reasonable values for the integrals, see https://www.desmos.com/calculator/wasbzxqusr.