Question

The partial fractions for

a²/(x(x²+a²)) are

Answer 1

$$\dfrac{a^2}{x(x^2+a^2)}=\dfrac{A}{x}+\dfrac{Bx+C}{x^2+a^2}$$

$$=\dfrac{Ax^2+Aa^2+Bx^2+Cx}{x(x^2+a^2)}$$

$$x^2: A+B=0$$

$$x^1: C=0$$

$$x^0: Aa^2=a^2$$

Then $A=1, B=-1, C=0.$

The partial fractions for $\dfrac{a^2}{x(x^2+a^2)}$ are

$$\dfrac{a^2}{x(x^2+a^2)}=\dfrac{1}{x}-\dfrac{x}{x^2+a^2}$$