Kalkylator för uppdelning i partialbråk

Your input: perform the partial fraction decomposition of $$$\frac{3 x^{2} + 2}{\left(x^{2} + 1\right)^{2}}$$$

The form of the partial fraction decomposition is

$$\frac{3 x^{2} + 2}{\left(x^{2} + 1\right)^{2}}=\frac{A x + B}{x^{2} + 1}+\frac{C x + D}{\left(x^{2} + 1\right)^{2}}$$

Write the right-hand side as a single fraction:

$$\frac{3 x^{2} + 2}{\left(x^{2} + 1\right)^{2}}=\frac{\left(x^{2} + 1\right) \left(A x + B\right) + C x + D}{\left(x^{2} + 1\right)^{2}}$$

The denominators are equal, so we require the equality of the numerators:

$$3 x^{2} + 2=\left(x^{2} + 1\right) \left(A x + B\right) + C x + D$$

Expand the right-hand side:

$$3 x^{2} + 2=x^{3} A + x^{2} B + x A + x C + B + D$$

Collect up the like terms:

$$3 x^{2} + 2=x^{3} A + x^{2} B + x \left(A + C\right) + B + D$$

The coefficients near the like terms should be equal, so the following system is obtained:

$$\begin{cases} A = 0\\B = 3\\A + C = 0\\B + D = 2 \end{cases}$$

Solving it (for steps, see system of equations calculator), we get that $$$A=0$$$, $$$B=3$$$, $$$C=0$$$, $$$D=-1$$$

Therefore,

$$\frac{3 x^{2} + 2}{\left(x^{2} + 1\right)^{2}}=\frac{3}{x^{2} + 1}+\frac{-1}{\left(x^{2} + 1\right)^{2}}$$

Answer: $$$\frac{3 x^{2} + 2}{\left(x^{2} + 1\right)^{2}}=\frac{3}{x^{2} + 1}+\frac{-1}{\left(x^{2} + 1\right)^{2}}$$$