Question

INTEGRATION sin(lnx)dx?

Answer 1

`int sin(lnx) dx = x/2 (sin(lnx) - cos(lnx))+C`

Explanation:

Integrate by parts:

`int sin(lnx) dx = int sin(lnx) d/dx (x) dx`

`int sin(lnx) dx = x sin(lnx) - int x d/dx (sin(lnx)) dx`

`int sin(lnx) dx = x sin(lnx) - int x cos(lnx)/x dx`

`int sin(lnx) dx = x sin(lnx) - int cos(lnx) dx`

and again:

`int sin(lnx) dx = x sin(lnx) - xcos(lnx) + int xd/dx(cos(lnx))dx`

`int sin(lnx) dx = x sin(lnx) - xcos(lnx) - int sin(lnx)dx`

The integral appears on both sides of the equation and we can solve for it:

`2int sin(lnx) dx = x sin(lnx) - xcos(lnx) +C`

`int sin(lnx) dx = x/2 (sin(lnx) - cos(lnx))+C`