Integrate sqrt(u-2): Trig or Straight Away?

[DigiDigi]

I was doing a question using u substitution and at one point,I got sqrt(u-2). How do I integrate it further?

Can I integrate is straight away with sqrt(u-2)(2/3) as answer or do I have to use trigonometric substitution? How to do if it's trigonometric?

 

[arildno]

Don't bother about the trig substitution. It will hiss at you, just as a nasty and venomous serpent would do.

And if you feel uncertain about integrating straight away, just substitute with the new variable s=u-2, and see where that leads you.

 

[DigiDigi]

So,I don't have to use trigonometric substitution?

 

[arildno]

Why tread on a rattle snake?
Much easier to substitute v=u-2

 

[DigiDigi]

Can I just integrate straight away? I mean I already u-substitution already do I need to do another substitution?

 

[arildno]

You never "need" to do any type of substitution. Substitutions are made in order to simplify the problem, but if the problem is simple enough (i.e, you what the answer will be), then there is not much point in further simplification, is there.

However, you expressed an uncertainty as to whether it was "allowed" to integrate straight away. (Everything is allowed to do in maths, provided it is correct. )

In order to alleviate that uncertainty, I pointed out to you that you CAN make the u-2 substition, if that makes you feel safer in doing the right thing.

 

[Mark44]

To elaborate on what arildno said, if you make the substitution v = u - 2, then dv = du, so the integral changes like so:
$$ \int \sqrt{u - 2}du = \int \sqrt{v}dv$$

The latter integral is pretty easy. A trig substitution is not needed here, but is very useful if the quantity in the radical is a sum or difference of squares.