Question

how would you do the integral of the sin(square root x)

? I just can't seem to get started

Answer 1

Let y = sqrt x.

Then y^2 = x
and 2y dy = dx
That makes your integral the same as
the integral of 2y sin y dy
which can be solved using integration by parts.
The integral is 2 sin y - 2 y cos y
= 2 sin (sqrt x) - 2 sqrt x cos(sqrt x)