Fixing syntax errors and adding assumptions on your variables:

Assuming[Y > 0 && Y1 > 0,
    X = 2 π Y^2 Y1 Integrate[1/((Y^2+s)^2*Sqrt[(Y1^2+s)]), {s,0,∞}]
];
X //TeXForm

$\frac{2 \pi Y \operatorname{Y1} \left(Y \cos ^{-1}\left(\frac{\operatorname{Y1}}{Y}\right)-\operatorname{Y1} \sqrt{1-\frac{\operatorname{Y1}^2}{Y^2}}\right)}{\left(Y^2-\operatorname{Y1}^2\right)^{3/2}}$

Then, replace Y with 1 and take the limit as Y1 goes to $\infty$:

Limit[X /. Y->1, Y1->∞]

2 π

Fixing syntax errors and adding assumptions on your variables:

Assuming[Y > 0 && Y1 > 0,
    X = 2 π Y^2 Y1 Integrate[1/((Y^2+s)^2*Sqrt[(Y1^2+s)]), {s,0,∞}]
];
X //TeXForm

$\frac{2 \pi Y \operatorname{Y1} \left(Y \cos ^{-1}\left(\frac{\operatorname{Y1}}{Y}\right)-\operatorname{Y1} \sqrt{1-\frac{\operatorname{Y1}^2}{Y^2}}\right)}{\left(Y^2-\operatorname{Y1}^2\right)^{3/2}}$

Then, replace Y with 1 and take the limit as Y1 goes to $\infty$:

Limit[X /. Y->1, Y1->∞]

2 π

Update

If you're interested in the Y and Y1 dependence for large Y1, you could do:

Series[X,{Y1,Infinity,2},Assumptions->Y>0] //TeXForm

$2 \pi +\frac{2 \pi Y^2}{\operatorname{Y1}^2}+O\left(\left(\frac{1}{\operatorname{Y1}}\right)^3\right)$