Question #f9641

1 Answer
Feb 14, 2018

int\ cos(x)/(sin^2(x)+sin(x))\ "d"x=ln|sin(x)/(sin(x)+1)|+C

Explanation:

\ \ \ \ \ \ int\ cos(x)/(sin^2(x)+sin(x))\ "d"x

Substitute u=sin(x) and "d"u=cos(x)\ "d"x. This gives
=int\ ("d"u)/(u^2+u)
=int\ ("d"u)/(u(u+1))

Separate to partial fractions since 1/(u(u+1))=1/u-1/(u+1):
=int\ (1/u-1/(u+1))\ "d"u

=ln|u|-ln|u+1|+C

=ln|u/(u+1)|+C

Substitute back u=sin(x):

=ln|sin(x)/(sin(x)+1)|+C