If we had the following integral:

$$\int_{a}^{b} {\big(1+x^2 \big)^s} \space dx$$

Where $s$ is not given. Is there any general formula for this integration that works for all $s\in \mathbb{R}$?


The substitution $x = \tan \theta$ leads to $\int_{\arctan a}^{\arctan b} \cos^{-2s-2} \theta \, d\theta$. According to Wolfram Alpha, this can be integrated using a hypergeometric function.

Namely, $$\int (1 + x^2)^s \, dx = x\,{}_2F_1(1/2, -s, 3/2, -x^2) + C,$$ where ${}_2F_1$ is the function described here.


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