Use the following Substitution to evaluate the following integrals.

Question

anonymous · Answer

$$x \sqrt{2+x}dx  $$ Using $$u ^{2} = 2 + x$$

rogue · Answer

Actually, no, hold on.

rogue · Answer

I don't know why you would use that substituion, its much easier to just use u = 2 + x, not u^2 = 2 + x.

anonymous · Answer

This is what we have been given in our homework :(

anonymous · Answer

Thanks btw for your help! I understand I need to rearrange to get X then put back into the equation however I am stuck on this
$$u ^{2} - 2 \sqrt{2+u ^{2}-2} * 2u du  $$ (I think) I am not familiar on how one would find the correct substitution

rogue · Answer

$$ u^2 = 2 + x, u = \sqrt {2+x}, x = u^2 - 2, 2udu = dx$$$$x \sqrt{2+x}dx \rightarrow \int\limits 2(u^2 - 2) u^{.5} u du \rightarrow 2 \int\limits u^{3.5} - 2u^{1.5} du$$

rogue · Answer

Not pretty numbers, but its doable.

anonymous · Answer

I am not sure what you have done to get to the middle equation! Sorry for being pain.

rogue · Answer

Plug in all the substitutions.
x = u^2 - 2
sqrt (2 + x) = u^.5
dx = 2udu

x sqrt (2+x) dx ----> (u^2 - 2) * u^.5 * 2udu

anonymous · Answer

oh okay the ^.5 were confusing me. I sound so stupid! Thanks a lot man!

rogue · Answer

lol, you don't sound stupid ;)
Good luck :)