Integrate The following (x^6057 + x^4038 + x^2019)* [(2x^4038 + 3x^2019 + 6)^1/6]

can you please show that Lim x->0 ln(x) = -infinity by the epsilon delta definition thank! :)

I am currently a bit stuck trying to solve this problem only using the Lambert W function and elementary functions. I succeeded in solving simpler functions like y = x^x and y = x^(x^2), but I cannot seem to solve this one. Here is my current best attempt (based off of the steps for x^(x^2)): y = x^(x^x) ln(y) = x^x * ln(x) = x^x * ln(x^(x / x)) = x^x * (1/x * ln(x^x)) = 1/x * x^x * ln(x^x) x * ln(y) = x^x * ln(x^x) = ln(x^x) * x^x = ln(x^x) * e^(ln(x^x)) = x*ln(x) * e^(x*ln(x)) x*ln(x) = W(x * ln(y)) x*e^x = e^W(x * ln(y)) x = W(e^W(x * ln(y))) But this isn't particularly useful as x is still in the formula. I could solve it if I knew how to solve ln(y) = 1/x * x^x * ln(x^x). Can you please help me?

can anyone help me find the sum of this power series?