Home » Blog » Evaluate the integral of x2 / (1+x2)2

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

by RoRi

Evaluate the following integral

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

We can evaluate this using integration by parts. Let

    \begin{align*} u &= x & du &= dx \\ dv &= \frac{x}{(1+x^2)^2} & v &= \frac{-1}{2(1+x^2)}. \end{align*}

Here, to arrive at v we make the substitution s = (1+x^2) and ds = 2x \, dx in the following,

    \[ \int \frac{x}{(1+x^2)^2} = \frac{1}{2} \int \frac{ds}{s^2} = \frac{-1}{2s} = \frac{-1}{2(1+x^2)}. \]

So, completing the integration by parts we have

    \begin{align*} \int \frac{x^2}{(1+x^2)^2} \, dx &= -\frac{x}{2(1+x^2)} + \frac{1}{2} \int \frac{1}{1+x^2} \, dx \\ &= -\frac{x}{2(1+x^2)} + \frac{1}{2} \arctan x + C \\ &= \frac{1}{2} \left( \arctan x - \frac{x}{1+x^2} \right) + C. \end{align*}

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):