Question

What is the integral ln x^1/3 dx.

Answer 1

Hello,

$\int\limits{ln(x^\frac{1}{3}) } \, dx =\frac{1}{3}*\int\limits{ln(x) } \, dx \\\\ =\frac{1}{3}*(x*ln(x)-x)+C=\frac{x}{3}*(ln(x)-1)+C$
Th$\int\limits{ln(x) } \, dx=x*ln(x)- \int\limits{x* \frac{1}{x} \, dx \\\\ =x*ln(x) -x+C$

The last integration is made by "parties(in french)"

$\int\limits{ln(x)} \, dx =x*ln(x)- \int\limits{x* \frac{1}{x} } \, dx \\\\ =x*ln(x)-x+C$