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".