Calculus: Early Transcendentals, Chapter 7, 7.1, Section 7.1, Problem 42

We need to make a substitution then use integration by parts.
Let us make the substitution:
ln(x) = t, so:
x = e^t
therefore dx = e^t dt
so our equation can be changed. int sin(ln(x))dx = int e^t(sin(t)) dt
Now use integration by parts.
Let u = sin(t) and dv = e^tdt
du = cos(t) and v = e^t
int e^t sin(t)dt = e^tsin(t) - int e^tcos(t) dt
We will call that equation 1.
Now we need to evaluate that second integral with integration by parts again.
int e^t cos(t) dt = e^t cos(t) - int e^t (-sin(t))dt
int e^t cos(t) dt = e^t cos(t) + int e^t sin(t)dt
Now let us plug this result for int e^t cos(t) dt back into equation 1.
int e^t sin(t)dt = e^tsin(t) - (e^t cos(t) + int e^t sin(t) dt)
add int e^t sin(t) dt to both sides:
2 int e^t sin(t) dt = e^t sin(t) - e^t cos(t)
sub back in our original t = ln(x) or e^t = x.
2 int sin(ln(x)) dx = xsin(ln(x)) - xcos(ln(x))
divide both sides by 2 and add the constant of integration. And were done!!!!
int sin(ln(x)) dx = (xsin(ln(x)) - xcos(ln(x)))/2 + c