The given function is \( \displaystyle \frac{x^{2} – 1}{x^{2} + 1} \) and we need to compute the integral of this function with respect to \( x \).
Solution:
\( \begin{aligned} &\int \frac{x^{2} – 1}{x^{2} + 1} \, dx \\ &= \int \frac{(x^{2} + 1) – 2}{x^{2} + 1} \, dx \\ &= \int \frac{x^{2} + 1}{x^{2} + 1} – \frac{2}{x^{2} + 1} \, dx \\ &= \int \left(1 – \frac{2}{x^{2} + 1}\right) dx \\ &= \int 1 \, dx \, – 2 \int \frac{1}{x^{2} + 1} \, dx \\ &= x \, – 2 \tan^{-1} x + C, \quad \text{C is a constant} \\ \end{aligned} \)Thus, the integral of the given function is
\( \boxed{\int \frac{x^{2} – 1}{x^{2} + 1} \, dx = x \, – 2 \tan^{-1} x + C} \)Please let me know in the comments if you find any errors in this solution.