Question: > 
> Find the value of $\int \dfrac{x^2+1}{x^4+1}\mathrm dx$.
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$$\int \frac{x^2(1-\frac{1}{x^2})}{x^2(x^2+\frac{1}{x^2})}\mathrm dx$$
$$\int \frac{d(x+\frac{1}{x})}{(x+\frac{1}{x})^2-2}$$

How to integrate 2nd $$\int \frac{x^2(1-\frac{1}{x^2})}{x^2(x^2+\frac{1}{x^2})}\mathrm dx$$ equation? **Even, how they had found that $$\int \frac{x^2(1-\frac{1}{x^2})}{x^2(x^2+\frac{1}{x^2})}\mathrm dx =\int \frac{d(x+\frac{1}{x})}{(x+\frac{1}{x})^2-2}$$** Where $\mathrm dx$ had gone in second line? How $\mathrm d$ came left?

I am not doing integration by parts or, technics of integration. I am just doing some standard integrals. I found this kind of 3 problems. How can I solve that simple way?

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Replying to my title : I have read somewhere that they had differentiate inside function? Why they had? What's the reason of differentiating that?

$$\int dx= x +c$$
So, if we differentiate the inside function then, it cancels with integration. So, is it wise to differentiate inside function? While I am differentiate inside function than, I have add another integral, but, that didn't happen here. So, how it is correct?

Answer: 
First of all this is a very standard question in indefinite integrals and this method is the fastest way to solve it. To address your concern,$$\frac{d}{dx}(x+\frac{1}{x})=1-\frac{1}{x^2}$$ When you substitute this in the integral, $dx$ cancels out and hence they directly wrote $d(x+\frac{1}{x})$.  

Also to avoid all this, just assume $$x+\frac{1}{x}=t \Rightarrow (1-\frac{1}{x^2})dx=dt$$ An this can be further solved. Remember that whenever you face functions such as, $$\int \frac{x^2\pm 1}{x^4\pm kx^2+1}dx$$ always use the approach used in Original Post. As this method converts the question into an integrable function when you take out $x^2$ as common, you can write $x^2+\dfrac{1}{x^2}=(x+\dfrac{1}{x})^2-2$ and the derivative of $(x+\dfrac{1}{x})$ will be present in numerator and hence it is an integrable form which is $$\frac{dt}{t^2-2}$$
Hint :Use Partial Fration i.e. $${\int{\frac{x^{2} + 1}{x^{4} + 1} d x}} = {\int{\left(\frac{1}{2 \left(x^{2} + \sqrt{2} x + 1\right)} + \frac{1}{2 \left(x^{2} - \sqrt{2} x + 1\right)}\right)d x}}$$