The method of integration by parts is not suitable in this case, since you cannot identify two different functions in the integrand 2x*(x^2 + 1)^10.
Since the functions 2x and (x^2 + 1)^10 are two polinomial functions and since derivative of x^2 + 1 yields 2x, you may use change of variable, instead of integration by parts.
The answer above ilustrates the substitution method and you should consider this method as the only one possible in this case.
We have given
consider f(t)=1 and g(t)= t^10
That is: `1/11 (x^2+1) +C`