Integrating ln x by parts

[Guest jasmine]

Hi!

 

Can someone please explain to me how to integrate: ln4x+cos4x+3. I don't know how to integrate ln4x. Please show me the steps, and explain what integration by part means and how to use it. I have tried to look on it but I am still not clear.

 

Thanks

[matt grime]

If f and g are two differentiable functions then

 

d/dx{f(x)g(x)} = f'(x)g(x)+f(x)g'(x)

 

we also know that integrating "undoes" differentiating, so we get

 

 

f(x)g(x) = int ( f'g+fg')dx

 

or, rearranging

 

fg - int f'gdx = int fg'dx

 

so if we spot an integral of the form int uvdx, and let u=f, g'=v then we can use this identity ot replace it with anothe, hopefully simpler integral and an evaluation of fg.

 

here log(4x) = log4 +logx, so we just want to know how to integrate logx

 

let f=logx, g'=1, then int logx dx = xlogx - int x*(1/x)dx = xlogx -x