How do I do this integration by substitution?

[shreddinglicks]

Homework Statement

∫1/(3+((2x)^.5))dx

the answer should be ((2x)^.5) - 3ln(3+((2x)^.5)) + c

I keep getting ((2x)^.5) - ln(3+((2x)^.5)) + c

Homework Equations

∫1/(3+((2x)^.5))dx

The Attempt at a Solution

I did:

u = 3 + ((2x)^.5)
du = 1/((2x)^.5) dx

du((2x)^.5) = dx

so my integral is:

((2x)^.5) ∫ (1/u) du

from there I integrated and got:

((2x)^.5) - ln(3+((2x)^.5)) + c

Please explain this problem is driving me crazy.

 

[AlephZero]

I did:

u = 3 + ((2x)^.5)
du = 1/((2x)^.5) dx

du((2x)^.5) = dx

so my integral is:

((2x)^.5) ∫ (1/u) du

You can't take the ((2x)^.5) outside the integral. x is not a constant!

You substituted u = 3 + ((2x)^.5)
so ((2x)^.5) = u - 3

 

[shreddinglicks]

You can't take the ((2x)^.5) outside the integral. x is not a constant!

You substituted u = 3 + ((2x)^.5)
so ((2x)^.5) = u - 3

Ok, so I take the derivative of

((2x)^.5) = u - 3

which gives me

du = 1/((2x)^.5)

I'm a bit confused, could you please elaborate on the steps?

 

[AlephZero]

You were doing OK the first time, as far as
du((2x)^.5) = dx

To do the substitution you need to get rid of ALL the x's.
So need to find dx in terms of u and du, not x and du.
That gives you
du(u-3) = dx.

 

[shreddinglicks]

You were doing OK the first time, as far as
du((2x)^.5) = dx

To do the substitution you need to get rid of ALL the x's.
So need to find dx in terms of u and du, not x and du.
That gives you
du(u-3) = dx.

Please excuse me if I'm sounding stupid

So I do:

(u-3)^2 = 2x

I use chain rule on left side and get:

2(u-3) = 2

the 2's cancel and I get:

du(u-3) = dx

 

[AlephZero]

You can do it like that if you want, but you already found
du((2x)^.5) = dx
and you know that
((2x)^.5) = u - 3
so you can just substitute u-3 for ((2x)^.5)

Either way, now have got
du(u-3) = dx
you can substitute that into the integral and finish the question.

 

[shreddinglicks]

You can do it like that if you want, but you already found
du((2x)^.5) = dx
and you know that
((2x)^.5) = u - 3
so you can just substitute u-3 for ((2x)^.5)

Either way, now have got
du(u-3) = dx
you can substitute that into the integral and finish the question.

Thanks, I didn't see it that way till you showed me. I got the right answer now.

Thanks a bunch!