Definite Integral Problem up-to infinity

Question

I was trying to solve simple integration problem, integration x from 0-infinite. Is there any particular answer to the question from any other methods? My try is I've shown on picture below.

Answer 1

Your mistake: As you change your variable to $z$, the variable is not change back.

Remark: Rather than $90$, you might want to work with radian.

$$\lim_{M \to \infty} \int_0^M x \, dx= \lim_{M \to \infty}\frac{M^2}2 = \infty$$

Answer 2

I would re-interpret problem to be for $0<a$, what is the limit as $a$ goes to infinity of $\int_0^a xdx.$ When $0<a$, $\int_0^a xdx = \displaystyle\frac{a^2}{2}.$ Therefore, the (re-interpreted) problem is equivalent to asking: what is the limit as $a$ goes to infinity of $\displaystyle\frac{a^2}{2}.$ Obviously, as $a$ goes to infinity, $\displaystyle\frac{a^2}{2}$ goes to infinity.

This approach is very similar to Siong Thye Goh's approach, except that I scrap the whole polar coordinates / radians approach and just stay with the algebra.