Question

Using the U- Substitution u=sqrt(2x), integral form 2-8 dx/ sqrt(2x) + 1 is equivalent to ...

I'm not quite sure how to solve using u substitution.

Help please!

Answer 1

Does the integral look like this:
\[\Large \int_a^b \frac{dx}{\sqrt{2x}+1} \]

Answer 2

yes. lower bound is 2, upper bound is 8.

Answer 3

Alright, so lets try to apply the hint they are giving us:
\[\Large u=\sqrt{2x} \]
Such that, if you differentiate that:
\[\Large \frac{du}{dx}=\frac{1}{\sqrt{2x}}=\frac{1}{u} \]

Answer 4

what about the plus 1?

Answer 5

well it's not part of the recommended substitution, I was just following the hint they have given us in your problem. it doesn't include a one in the substitution.

Answer 6

Oh. ok.

Answer 7

But now you have carried out all the necessary substitutions, remember that it is essentially important that for every substitution you take, you also want to substitute the differential (in this case dx) back into the integral (original problem)

So if you agree with my steps above you will see that:
\[\Large dx=udu \]

Answer 8

ok.

Answer 9

Can you solve the integral from here?

Answer 10

I'm not sure what the next several steps are?

Answer 11

Substitute back:
\[\Large \int_a^b \frac{u}{u+1}du \]

Answer 12

you got to this point?

Answer 13

I integrate?

Answer 14

thanks.

Answer 15

well yes, you do, but basically you want to do a longhand division first, but then you can integrate.