Integrate by Parts: arctan(√x)dx Explained

[Aerospace93]

Homework Statement

∫arctan(√x)dx.

Using the substitution √x=t:
∫arctan(√x)dx = ∫arctan(t)dt²

This is what I've got written in a solution manual. I don't see why the dt would be squared. Could anyone care explaining me? thanks

 

[HallsofIvy]

The substitution t= \sqrt{x} is the same (for positive x and t) as t^2= x so that dx= d(t^2)= 2tdt.

What is done here is "integration by substitution". The result is 2\int arctan(t) tdt which can now be done by "integration by parts".