Question

What is `int ln(2x)^2dx`?

Answer 1

`=2x\ln(2 x) -2x +C`

Explanation:

Given that

`\int ln(2x)^2 \ \d x`

`=\int 2ln(2x) \ \d x`

`=2\int (lnx+\ln 2) \ \d x`

`=2\int lnx\ dx+2\int \ln 2 \d x`

`=2(\ln x \int 1\ dx-\int (1/dx(lnx)\cdot \int1 \ dx) dx)+2\ln 2\int 1 \d x`

`=2(x\ln x -\int (1/x\cdot x) dx)+2\ln 2 \ (x)+C`

`=2(x\ln x -x)+2x\ln 2 +C`

`=2x\ln(2 x) -2x +C`

Answer 2

The answer is `=2xln2x-2x+C`

Explanation:

A slightly different method

The integral is

`I=intln(2x)^2dx=2intln2xdx`

Perforn an integration by parts

`intuv'dx=uv-intu'vdx`

`u=ln2x`, `=>`, `u'=1/2x*2=1/x`

`v'=1`, `=>`, `v=x`

Therefore,

`I=2xln2x-2int1/x*xdx`

`=2xln2x-2intdx`

`=2xln2x-2x+C`