Question

exFormula1" title="integration \: of \frac{1}{2} ln(2x {^{2} ) " alt="integration \: of \frac{1}{2} ln(2x {^{2} ) " align="absmiddle" class="latex-formula">

Answer 1

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

$\dfrac12\ln(2x^2)=\dfrac{\ln2+\ln x^2}2=\dfrac{\ln2+2\ln x}2=\ln\sqrt2+\ln x$

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

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