integrate a variable '$u$' with a '$dx$', is that possible? (Basic Integrals)

Question

1. Are these integration written correctly?
2. is it possible to do integration like second one with the respect to $x$?

First: $$ \int (1+\tan^2(ax))\,dx = \frac 1a \tan(ax) +c. $$ Second (This is not $u$-Substitution): $$ \int u’(1+\tan^2u) \,dx = \tan u +c. $$

Answer 1

I'm not sure what you mean by "written correctly", but the solution for the first integral is correct. The second integral is also correct, but despite your claim, this is in fact $u$-substitution. When calculating integrals in practice, you may use the fact that $u'dx=du$, and so the integral can be rewritten entirely in terms of $u$; this is precisely what is meant by $u$-substitution.

Answer 2

The expression $u'\,dx$ is the same thing as $\dfrac{du}{dx}\,dx,$ so it's the same as $du.$

$$ \int u'(1+\tan^2u) \,dx = \int(1+\tan^2u)\,du $$

However, if $u = ax,$ then $\dfrac {du}{dx} = a,$ so $dx = \dfrac{du} a, $ and you have \begin{align} & \int (1+\tan^2(ax)) \, dx = \int\frac{u'} a(1+ \tan^2 u) \, dx \\[15pt] = {} & \frac 1 a \int(1+\tan^2u)\, du = \frac 1 a \tan u + C = \frac 1 a \tan(ax) + C. \end{align}