Ambiguous integral What is the real integral of the function $$f(x) =\frac{1-x^2}{(1 + x^2)^2}$$? Is it $F_1(x) = \frac{x}{1 + x^2} + C$ or $F_2(x) = \arctan x + C$ ? The brochure I was reading gave the first result...--prophetes.ai

Ambiguous integral What is the real integral of the function $$f(x) =\frac{1-x^2}{(1 + x^2)^2}$$? Is it $F_1(x) = \frac{x}{1 + x^2} + C$ or $F_2(x) = \arctan x + C$ ? The brochure I was reading gave the first result straightaway without any intermediate steps. I went to wolframalpha to try to see what where the steps and to my surprise wolframalpha has derived the second function%20%2F%20\(x%5E2%20%2B%201\)%20%5E%202%7D). * * * I'm terribly sorry. I am preparing for exam non-stop for several days and lost my focus. I probably should go and have some rest. Am I required to remove this questions?

HINT:

$$\dfrac{1-x^2}{(1+x^2)^2}=\dfrac{\dfrac1{x^2}-1}{\left(x+\dfrac1x\right)^2}$$

Now $\displaystyle\int\left(\dfrac1{x^2}-1\right)dx=?$