What is the temperature profile in 1D heat transfer(steady, no heat generation) and why is it linear? Prove.

The following figure shows a heat transfer through a plane wall,

The temperature profile in 1D heat transfer is linear. It is nothing but just a slope dT/dX.

For one-dimensional heat transfer, dT/dx can be replaced by,

Also, we know the heat diffusion conduction equation for steady, 1D equation without heat generation is given by

If this equation is integrated 2 times we get a temperature distribution through the wall.

The above equation is of the form

which is nothing but an equation of a straight line with slope m.