Every solution I've seen uses trig sub by letting $u=\tan (x)$.
Why can't I just let $u=x^2+1$?
Then $\dfrac{du}{dx}=2x$, then $dx=\dfrac{1}{2x}du$, so$$\int \sqrt{x^2+1}\,dx=\frac{1}{2x}\int \sqrt{u}\,du,$$so finally you get $\dfrac{(x^2+1)^\frac{3}{2}}{12x}+C$.
Obviously this is wrong. But why? Can you just not substitute a function like $f(x)=x^2+1$ as $u$? I'm sure I'm making a mistake somewhere? Sorry in advance if this is obvious.