`int_(pi/4)^(pi/2) cot^3(x) dx` Evaluate the integral

`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))`

`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))`