Ambiguous integral

Question

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.

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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?

Answer 1

HINT:

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

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

Answer 2

Differentiate both. You will see $F_1$ works and $F_2$ does not.