Question

Why does integral of sin(sqrtx)dx has to be done with integration by parts and not just substitution or advanced guessing?

Answer 1

\[u=\sqrt x~~\Rightarrow~~du=\frac{1}{2\sqrt x}~dx~~\iff~~2u~du=dx\]
So your integral can be written as
\[\int\sin\sqrt x~dx=2\int u\sin u~du\]
the latter which requires integration by parts.