Which Integration Technique Should I Use for This Triple Integral?

[MathematicalPhysics]

$$\iiint {\sqrt(R^2 - 2aR\cos\theta + a^2)} R^2 \sin\theta\,dR\,d\theta\,d\phi$$

with the integration over R between 0 and a
the integration over between 0 and pi
the integration over between 0 and 2pi

Should I use integration by parts or should I take the R^2 sin(theta) under the square root?

Any hints and tips are much appreciated!

 

[arildno]

Do the $$\theta$$ integration first (note that the sine outside the root makes this easy).

Beware absolute values in your R integration!

 

[MathematicalPhysics]

Do I need to change the order of integration then and have new limits or can I choose to rearrange it to a more convenient form, like

$$\iiint {\sqrt(R^2 - 2aR\cos\theta + a^2)} R^2 sin\theta\,d\theta\,dR,d\phi$$

then integrating wrt $$\theta$$ by parts?

 

[arildno]

You don't do it by parts!

Note that your integrand equals:
$$\frac{\partial}{\partial\theta}\frac{R}{3a}(R^{2}-2aR\cos\theta+a^{2})^{\frac{3}{2}}$$