[tex]\displaystyle\int\frac{\ln x^7}x\,\mathrm dx=7\int\frac{\ln x}x\,\mathrm dx[/tex]
Set [tex]y=\ln x[/tex], recalling that [tex]\mathrm dy=\dfrac{\mathrm dx}x[/tex], so the integral is
[tex]\displaystyle7\int y\,\mathrm dy=\dfrac72y^2+C=\dfrac72(\ln x)^2+C[/tex]
