Question

How do you integrate `int cos(lne(x))` using integration by parts?

Answer 1

`intcos(ln(x))dx=x(cos(ln(x))-sin(ln(x)))/2+C`, `C in RR`

Explanation:

`intcos(ln(x))dx`
let : `X=ln(x)`
`x=e^X`
`dx=e^XdX`
so :
`intcos(ln(x))dx=intcos(X)e^XdX`
Using integration by parts :
`intuv'dX=uv-intu'vdX`
there :
`u=e^X` `v'=cos(X)`

`u'=e^X` `v=sin(X)`

So : `intcos(X)e^XdX=(-sin(X)e^X)-int(sin(X)e^X)dx`

`=-sin(X)e^X-intsin(X)e^XdX`

Using integration by parts again :

`intsin(X)e^XdX=-cos(X)e^X+int(cos(X)e^XdX`

So :

`intcos(X)e^XdX=-sin(X)e^X+cos(X)e^X-int(cos(X)e^XdX`

`2intcos(X)e^XdX=e^(X)(cos(X)-sin(X))`

`intcos(X)e^XdX=(e^(X)(cos(X)-sin(X)))/2+C`, `C in RR`

`intcos(ln(x))dx=cancel(e^(ln(x)))^(=x)(cos(ln(x))-sin(ln(x)))/2+C`, `C in RR`

`intcos(ln(x))dx=x*(cos(ln(x))-sin(ln(x)))/2+C`, `C in RR`

\0/ here's our answer !