Question

`integration \: of (1)/(2) ln(2x {^(2) )`

Answer 1

Using properties of the logarithm, you can rearrange the integrand as

`\frac12\ln(2x^2)=\frac{\ln2+\ln x^2}2=\frac{\ln2+2\ln x}2=\ln\sqrt2+\ln x`

Then recall that the integral of
`\ln x` is

`\displaystyle\int\ln x\,\mathrm dx=x(\ln x-1)+C`

(or you can find that out by integrating by parts) and so

`\displaystyle\int\frac12\ln(2x^2)\,\mathrm dx=(\ln\sqrt2)x+x(\ln x-1)+C`