`int_36^81(sqrt x -5)^(3/2)/sqrt x dx=?`

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`int_36^81(sqrt x -5)^(3/2)/sqrt x dx=|(t=sqrt x-5),(dt=dx/(2 sqrt x) => 2dt=dx/sqrt x),(t_1=sqrt36-5=1),(t_2=sqrt81-5=4)|=`

We have made substitution `t=sqrt x-5` and `t_1` and `t_2` are now new limits of integration for variable `t.` In the following line we use the fact that `int x^(3/2)dx=2/5 x^(5/2)`.

`2int_1^4t^(3/2)dt=2 cdot 2/5t^(5/2)|_1^4=4/5(4^(5/2)-1^5/2)= 4/5(32-1)=124/5` ` `

So your solution is:  `int_36^81(sqrt x -5)^(3/2)/sqrt x dx=124/5`. 

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