Sunday, February 10, 2013

`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))`

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