What is int (ln(sinx))*cosxdx?

1 Answer
Nov 8, 2015

sin(x)ln(sin(x))-sin(x)+C
or
sin(x)(ln(sin(x))-1)+C

Explanation:

int(ln(sin(x))*cosxdx

let s = sin(x) (normally I would use "u", but we will use it later)
then ds = cos(x)dx

now we have
intln(s)*ds

To integrate this we use integration by parts
int udv = uv - int vdu

let u = ln(s) let dv = ds
so du = 1/sds and v=s

the integral can now be rewritten as
ln(s)s- int s*1/sds
sln(s)- int 1ds
sln(s)- s +C

Now substitute sin(x) for s

sin(x)ln(sin(x))-sin(x)+C
or
sin(x)(ln(sin(x))-1)+C