Calculus: Early Transcendentals, Chapter 7, 7.2, Section 7.2, Problem 36

int_(pi/4)^(pi/2)cot^3(x)
Let's evaluate the indefinite integral by rewriting the integrand as,
intcot^3(x)=intcot(x)cot^2(x)dx
Now use the identity:cot^2(x)=csc^2(x)-1
=intcot(x)(csc^2(x)-1)dx
=int(cot(x)csc^2(x)-cot(x))dx
=intcot(x)csc^2(x)dx-intcot(x)dx
Now let's evaluate intcot(x)csc^2(x)dx by integral substitution,
Let u=cot(x)
=>du=-csc^2(x)dx
intcot(x)csc^2(x)dx=intu(-du)
=-intudu
=-u^2/2
substitute back u=cot(x)
=-1/2cot^2(x)
Use the common integral intcot(x)dx=ln|sin(x)|
:.intcot^3(x)dx=-1/2cot^2(x)-ln|sin(x)|+C , C is a constant
Now let' evaluate the definite integral,
int_(pi/4)^(pi/2)cot^3(x)dx=[-1/2cot^2(x)-ln|sin(x)|}_(pi/4)^(pi/2)
=[-1/2cot^2(pi/2)-ln|sin(pi/2)|]-[-1/2cot^2(pi/4)-ln|sin(pi/4)|]

=[-1/2*0-ln(1)]-[-1/2(1)^2-ln(1/sqrt(2))]
=[0]+1/2+ln(1/sqrt(2))
=1/2+ln(2^(-1/2))
=1/2-1/2ln(2)
=1/2(1-ln(2))