Integration ambiguity

Question

I am trying to Evaluate $$I=\int\frac{x^4+1}{x^6+1}dx$$ Now $I$ can be written as

$$I=\int\frac{x^4(1+\frac{1}{x^4})}{x^6(1+\frac{1}{x^6})}dx=\int\frac{(1+\frac{1}{x^4})}{x^2(1+\frac{1}{x^6})}dx$$

Now i used the Substitution $$\frac{1}{x}=t$$ $\implies$ $\frac{dx}{x^2}=-dt$

So $I$ becomes

$$I=-\int\frac{1+t^4}{1+t^6}dt$$ and Since $t$ is dummy variable we have

$$I=-I$$ so $$I=0$$ I could not find where is the flaw in my solution.

Answer 1

The step where you conclude $I=-I$ is invalid.

Suppose $\int\frac{x^4+1}{x^6+1}dx=f(x)$

Then you can only conclude $f(x)=-f(t)$ which implies $f(x)=-f({1\over x})$ and indeed $f(1)=f(-1)=0$ but you cannot say anything more than that.

Answer 2

Had you used limits of integration, you would have noticed the difference. The integration constant won't be the same, would it?