$\int_{0}^{\infty}{\ln(x+3){x^{-2}} \mathrm d x}$ converges?

Question

I ran into this question:

show convergence/divergence of:

$$\int_{0}^{\infty}{\ln(x+3){x^{-2}} \mathrm d x}$$

I tried for a long time and I'm kind'a lost. according to the answer, it should diverge.

Thanks in advance,

yaron.

Answer 1

Substitute $u=\ln{x}+3$.

$\displaystyle \int_{3}^{\infty} ue^{2(3-u)}e^{u-3}\mathrm{d}u=\int_{3}^{\infty} ue^{3-u}\mathrm{d}u=\left[ -ue^{3-u}-e^{3-u}\right]_3^{\infty}$ which diverges for $u=3$.

P.S. Happy $\tau$ day!