In the context of a physics calculation, I have encountered an integral of the form:
$$\int d^3k_1d^3k_2F(\vec{k_1},\vec{k_2})$$
The notes that I'm reading tell me that I need to symmetrize the expression of $F$ over the momenta $\vec{k_1}$ and $\vec{k_2}$, because, as the integrals are symmetric in these arguments, the antisymmetric part will vanish. Now, the expression of $F$ is quite complex, but it's entirely symmetric except for the term:
$$\dfrac{\vec{k_1}\cdot(\vec{k_1}+\vec{k_2})}{k_1^2}=1+\dfrac{\vec{k_1}\cdot \vec{k_2}}{k_1^2}$$
What I don't understand is how I can symmetrize this fraction. I have thought of writing:
$$\vec{k_2}=\dfrac{1}{2}(\vec{k_1}+\vec{k_2})+\dfrac{1}{2}(\vec{k_2}-\vec{k_1})$$
but the presence of $k_1^2$ in the denominator of the original expression I want to symmetrize seems to complicate things, and I cannot find a way to separate it as the sum of a symmetric and an antisymmetric part.
Any help would be greatly appreciated!