Sunday, January 24, 2016

`int_4^9 ln(y)/sqrt(y) dy` Evaluate the integral

`int_4^9 lny/sqrty dy`


To evaluate, apply integration by parts `intudv=uv-intvdu` .


So let:


`u = ln y `    


and    


`dv=int1/sqrty dy`


Then, differentiate u and integrate dv.


`du=1/y dy`


and


`v=int 1/sqrty dy = int y^(-1/2)=2y^(1/2)`


Plug-in them to the formula. So the integral becomes:


`int lny/(sqrty)dy`


`= lny*2y^(1/2) - 2y^(1/2)*1/ydy`


`=2y^(1/2)lny - 2int y^(-1/2) dy`


`=2y^(1/2)lny-2*2y^(1/2)`


`=2sqrtylny - 4sqrty`


And, substitute the limits of the integral.


`int _4^9 lny/ydy`


`=...

`int_4^9 lny/sqrty dy`


To evaluate, apply integration by parts `intudv=uv-intvdu` .


So let:


`u = ln y `    


and    


`dv=int1/sqrty dy`


Then, differentiate u and integrate dv.


`du=1/y dy`


and


`v=int 1/sqrty dy = int y^(-1/2)=2y^(1/2)`


Plug-in them to the formula. So the integral becomes:


`int lny/(sqrty)dy`


`= lny*2y^(1/2) - 2y^(1/2)*1/ydy`


`=2y^(1/2)lny - 2int y^(-1/2) dy`


`=2y^(1/2)lny-2*2y^(1/2)`


`=2sqrtylny - 4sqrty`


And, substitute the limits of the integral.


`int _4^9 lny/ydy`


`= (2sqrtylny - 4sqrty)|_4^9`


`=(2sqrt9 ln9 - 4sqrt9)-(2sqrt4ln4-4sqrt4)`


`=(2*3ln9 - 4*3)-(2*2ln4-4*2)`


`=(6ln9-12)-(4ln4-8)`


`=6ln9-4ln4-4`


`=ln(9^6)-ln(4^4)-4`


`=ln (9^6/4^4)-4`


`=ln(((3^2)^6)/(4^4))-4`


`=ln (((3^3)^4)/(4^4))-4`


`=ln((3^3/4)^4)-4`


`=4ln(27/4)-4`



Therefore, `int_4^9 lny/sqrtydy = 4ln (27/4)-4` .

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