$u$-substitution into integral For example we want to integrate  following integral
$$\int_0^4 2x\cdot((9+x^2)^{1/2})\,dx$$
I have read that if we denote $u$ as  $u=9+x^2$ then  $du=2x*dx$  everything is clear here but then there was used such kind of method: the author simply  put $x=0$ and $x=4$ into the original $u$ definiton  got $u=9$ and $u=25$  and   finally wrote integral as $$\int_{u=9}^{u=25} u^{1/2}\,du.$$
My question is: is it correct? Or if I will use the same method to solve   similar integrals next time, will I be wrong or not? thanks

It is correct, because you have the following:
$$\int_{a}^{b} f(\varphi(t)) \cdot \varphi'(t)\,\mathrm{d}t = \int_{\varphi(a)}^{\varphi(b)} f(x)\,\mathrm{d}x$$
Hope this helps.
