2. int (2x^2dx)/(sqrt (x^3)+1)=

(6*arcSinh[n])/sqrt[3] - 2sqrt[3]*n ------------------------------------------------The math equation used is based on LaTeX syntax, to view properly kindly use a LaTeX reader.

Let's assess the form of the given integral. We have ∫2x^2dx/(√x^3+1),which appears to be of the Lazy Man's form ∫ udv/(v^2), where u is the function with the variables, dv is the differential removed and as in our function (x^3+1).Let's proceed to use this information to perform the integral. Solving: first, rewrite the integral as ∫2x^2dx/(x^(3/2)+1)^(1/2). From our observation, here, let's say u(x) = 2*x^2 and dv = dx then dv is equal to x^(-1/2) and we can substitue dv as dx and sqrt(x^3 + 1) is considered as v^2 where the derivative of v(v(x)) would be x^2. Substituting we get The integral can be solved using a u-substitution method setting u equal to the denominator such as u=x^3+1; then x^2dx becomes du/3.Then let's apply the "Lazy Man's" integral technique, which sets our equation 1/v*integral(u*dv).With integration, we can then restate the solved equation for easier visual appeal! This integral would require use of antidifferentiation techniques with cancelling out fraction parts.