`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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