Integration by parts for definite integrals

Integration by parts states $$\int_2^4 uv' dx = uv\big|_2^4 - \int_2^4 u'v dx$$

We have $uv=1$. Hence $uv\big|_2^4 = 0$.

The equation becomes $$\int_{2}^{4} \frac {dx} {x \ln x} = 0 + \int_{2}^{4} \frac {dx} {x \ln x}$$

and no conclusion can be drawn.

$$I=\int_2^4\frac{dx}{x\ln x}$$ $$u=\ln(x)\Rightarrow dx=xdu,\,u\in[\ln(2),\ln(4)]$$ $$I=\int_{\ln(2)}^{\ln(4)}\frac{xdu}{xu}=\int_{\ln(2)}^{\ln(4)}\frac{1}{u}du=\left[\ln(u)\right]_{\ln2}^{\ln4}=\ln(\ln4)-\ln(\ln2)=\ln(2\ln2)-\ln(\ln2)$$ $$=\ln(2)+\ln(\ln2)-\ln(\ln2)=\ln(2)$$ This is really the easiest way to do it, no conclusion can be drawn purely from IBP

We have $$\int_{2}^{4}\frac{dx}{x\ln(x)}=\big[1\big]_{x=2}^{x=4}+\int_{2}^{4}\frac{dx}{x\ln(x)}=1-1+\int_{2}^{4}\frac{dx}{x\ln(x)}=\int_{2}^{4}\frac{dx}{x\ln(x)}$$ so we don't get anywhere using integration by parts.