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Evaluate the integral of 1 / (x (x2 + 1)2)

by RoRi

Compute the following integral.

    \[ \int \frac{1}{x(x^2+1)^2} \, dx. \]

To evaluate this we use partial fractions. First, we write

    \[ \frac{1}{x(x^2+1)^2} = \frac{A}{x} + \frac{Bx+C}{x^2+1} + \frac{Dx + E}{(x^2+1)^2}. \]

This gives us the equation

    \[ A(x^2+1)^2 + (Bx+C)(x)(x^2+1) + (Dx + E)x = 1. \]

First, we evaluate at x = 0 to find that A = 1. Using this value of A we obtain the equation

    \[ x^4 + 2x^2 + 1 + Bx^4 + Bx^2 + Cx^3 + Cx + Dx^2 +Ex = 1. \]

Equating like powers of x this gives us the system of equations:

    \begin{align*} B+1 &= 0 \\ C &= 0 \\ 2+B+D &= 0 \\ C+E &= 0. \end{align*}

We solve this system to obtain,

    \[ B = -1, \quad C = 0, \quad D = -1, \quad E = 0. \]

Therefore,

    \begin{align*} \int \frac{dx}{x(x^2+1)^2} &= \int \left( \frac{A}{x} + \frac{Bx+C}{x^2+1} + \frac{Dx+E}{(x^2+1)^2} \right) \, dx \\ &= \int \frac{dx}{x} - \int \frac{x \, dx}{x^2+1} - \int \frac{x \, dx}{(x^2+1)^2} \\ &= \log|x| - \frac{1}{2} \log (x^2+1) + \frac{1}{2(x^2+1)} + C. \end{align*}

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