Help with integrating arctan function

[ssmith147]

Homework Statement

INTEGRATE dx / (2 * root(x)) * (1 + x)

Homework Equations

That's pretty much it!

The Attempt at a Solution

I received this question on a Calc II exam, so I'm only looking for the solution for my own understanding (I'm sure I already got it wrong). My instinct is that this is an arctan integral in the form of 1 / 1 + u^2. Unfortunately, after playing around with it for about 30 minutes I was unable to find a way to get the denominator into the form 1 + u^2.

Am I missing something or did I misjudge the solution?

Anyw help would be greatly appreciated!

 

[p21bass]

What if you use $$\sqrt{x}$$ as u?

What does du/dx become, and then the rest of the equation?

 

[ssmith147]

u = root(x) so du = 1/(2*root(x)).

INT [1 / ( 2 * root(x))] * [1 / (1 + x)] dx
= INT [1 / (1 + x)] du = ln|1 + x|

That doesn't seem to work, though. If I differentiate this function I only get the original function with respect to u, not with respect to x since the value of u is eliminated by the value of du during the substitution. I've tried playing with this but I'm not seeing a way to successfully balance the value of du and retain u in the function.

Am I missing the obvious here?

 

[Bohrok]

You need to express ∫[1 / (1 + x)] du in terms of u with no x at all since you have to integrate with respect to u. So using the substitution u = √x → x = u², then that gives you ∫du/(1 + u²) which you can now integrate correctly.

 

[ssmith147]

Wow, I can't believe I didn't see that. During the exam I tried using root(x) for u but I never thought about root(x)^2 = x. Tunnel vision must have set in- I should have seen that!

So the answer would be arctan(root(x))- I suppose I can console myself knowing I saw arctan(x) correctly. Devil's in the details, unfortunately.

I really appreciate the feedback on this. At least I know what I did wrong now!

 

[p21bass]

That's it. Knowing (or figuring out) what to make u in these cases can be a real pain in the neck. And in some cases - such as this - it may not be particularly obvious whether simple substitution or integration by parts is necessary.

I was out last night, so thanks to Bohrok for explaining it more thoroughly.