Triple Integral with Spherical Coordinates

[veritaserum]

Homework Statement

Evaluate $$\int\int\int$$ 1/$$\sqrt{x^{2}+y^{2}+z^{2}+3}$$ over boundary B, where B is the ball of radius 2 centered at the origin.

Homework Equations

Using spherical coordinates:
x=psin$$\Phi$$cos$$\Theta$$
y=psin$$\Phi$$sin$$\Theta$$
z=pcos$$\Phi$$

Integral limits:
dp - [0,2]
d$$\Phi$$ - [0,$$\pi$$]
d$$\Theta$$ - [0,2$$\pi$$]

The Attempt at a Solution

I am just having trouble finding a good substitution for the integrand. When I substitute x,y, and z with the spherical substitutions, I just get a huge jumbled mess that I can't make any sense of.

 

[veritaserum]

whoops sorry, the integrand should be
$$\frac{dV}{\sqrt{x^{2}+y^{2}+z^{2}+3}}$$

 

[gabbagabbahey]

When I substitute x,y, and z with the spherical substitutions, I just get a huge jumbled mess that I can't make any sense of.

Show us. And keep in mind that $\sin^2\eta+\cos^2\eta=1$.

 

[veritaserum]

lol even trying to type that out is a huge jumbled mess in itself. i understand the property you gave me, it's just that i am not able to factor out enough terms such that i leave that identity intact.