I have been doing integration by substitution. And I wonder whether there is a general method or approach, as to what I can pick as $u$. I have had these specific tasks, were the teacher gave a hint as to what $u$ should be and the rest we had to solve ourselves:
$\int\ln(1+\sqrt{x})\,dx\quad$ Hint: $u=1+\sqrt{x}$.
$\int \frac{e^{\sqrt{x}}}{\sqrt{x(1+e^{\sqrt{x}})}}\,dx\quad$, Hint: $u=1+e^{\sqrt{x}}$
$\int \frac{x}{\sqrt{a^2-x^2}}\,dx\quad$ Hint: $u=a^2-x^2$
But how can he see what should be picked as $u$? My experience is that what is picked for $u$ is that what is the least complex. Which means that picking the entire denominator for $u=\sqrt{x(1+e^{\sqrt{x}})}$ in (2) and the same for (3) $u=√a^2-x^2$, would lead nowhere or would be too complex. Or do you disagree would it be just as good picking the entire denominator in (2) and (3)?
Thank you.
Too long for a comment.
u substitution is essentially a reverse chain rule. The change rule allows you to take the derivative of a composite function with respect to its argument, then multiply by the derivative of the argument with respect to $x$. u sub does something similar. So you want to focus on i) the argument of a composite function ii) with an eye out for a manipulation of the argument that already has a factor of its derivative in the integrand.
For problem 1, $u=1+\sqrt{x}$. There's no opportunity for part of the argument's derivative to appear in the integrand.
For problem 2, there are multiple possibilities based on that principle. Note $1/\sqrt{x}$ is a factor of the integrand and $\sqrt{x}$ occurs as an argument multiple times. $(\sqrt{x})'=1/2\sqrt{x}$ So $u=\sqrt{x}$ is a good choice, though not the best.
$\frac{e^{\sqrt{x}}dx}{\sqrt{x(1+e^{\sqrt{x}})}}=\frac{e^udu}{2\sqrt{1+e^u}}$
Then you can apply the same principle again. The only composite function now is $\sqrt{1+e^u}$ and conveniently its derivative is a factor of the integrand already in place. This suggests we let $v=1+e^u\implies dv=e^udu$
$\frac{e^udu}{2\sqrt{1+e^u}}=\frac{dv}{2\sqrt{v}}$
Using $v=1+e^{\sqrt{x}}$ would have cut to the chase, but either way is pretty quick.
Problem 3 is pretty clear. There's only one composite function and a scalar multiple of its derivative is in the integrand.
In general, substitution takes $\int f(g(x))g'(x) dx$ to $\int f(u)du$ via $u=g(x)$.
So good rule of thumb focus on arguments of the composite function and multiples of its derivative.