Answer
$A = \frac{1}{2}; B = \frac{1}{2}$
Work Step by Step
The given equation is:
$$\dfrac{x}{(x-a)(x+a)} =\dfrac{A}{(x-a)}+\dfrac{B}{(x+a)} ~~~(1)$$
On multiplying equation (1) with $(x-a) (x+a)$, we obtain:
$$x =A(x+a)+B(x-a)~~~(2)$$
Now, we will put $x=a$ in equation (2) to eliminate $B$:
$$a=A(a+a)\Rightarrow A=\dfrac{1}{2}$$
Next, we will put $x=-a$ in equation (2) to eliminate $A$:
$$-a=B(-a-a)\Rightarrow B=\dfrac{1}{2}$$
We got:
$$A = \dfrac{1}{2}; B = \dfrac{1}{2}$$
