Hi I'm following an Analysis course and one of the exercises is to prove that:

1. $$\int\_{1/2}^1 x^n \log x dx \approx -1/n^2 $$
2. $$\left|\int\_0^{1/2} x^n \log x dx\right| = O\left((1/2)^n\right)$$

At first I thought to solve this by the laplace method and write the integrand as a power of $e$ and then simplified by Taylor series. 

Is there someone who can help me so that I will understand how to solve this asymptotic approximation.
Many thanks!!