Calculus: Early Transcendentals, Chapter 7, 7.2, Section 7.2, Problem 18

intcot^5(theta)sin^4(theta)dTheta=
int[(cos^5(theta))/(sin^5(theta))]sint^4(theta)dTheta=
int[(cos^5(theta))/(sin^(theta))]d(Theta)=
intcos^4(theta)[cos(theta)/sin(theta)]d(Theta)=
int(cos^2(theta))^2[cos(theta)/sin(theta)]d(Theta)=
int(1-sin^2(theta))^2[cos(theta)/sin(theta)]d(Theta)=
Integrate using u-subsitution.
Let
u=sin(theta)
(du)/[d(theta)]=cos(theta)
d(theta)=(du)/[cos(theta)]

int(1-u^2)^2[cos(theta)/u][(du)/(cos(theta))]=
int[1-2u^2+u^4]/udu=
int[(1/u)-2u+u^3]du=
ln|u|-(2u^2)/2+u^4/4+C=
ln|u|-u^2+1/4u^4+C=
ln|sin(theta)|-sin^2(theta)+1/4sin^4(theta)+C

The final answer is: ln|sin(theta)|-sin^2(theta)+1/4sin^4(theta)+C